mscroggs.co.uk Blog & Puzzles

World Cup stickers 2026

02 May 2026 12:00:00 GMT

The men's World Cup is coming up, so it's time to work out the cost of filling the official Panini sticker book again.

Panini are boasting that this year's sticker book contains more stickers than ever, which can only mean one thing: you're going to have to buy more stickers than ever.But how much more than last time is completing this sticker book going to cost you?

Buying stickers

This year, you can buy:

A starter pack that includes a sticker book with 6 stickers (that are always the same and hence guaranteed to contain no duplicates) and 4 packs of 7 random stickers. This costs £4.99.
Packs of 7 random stickers. Various sources claim these are £1.25 each, although our local Morrison's is selling them for £2.
Multipacks of 6 or 8 packs of 7 random stickers. Both these multipacks work out at £1.25 per pack.
Multipacks of stickers that come with a nice metal box to keep your swaps in (you'll need a few of these if you're planning to complete your collection). These arereleasing later than the rest of the collection, but look like they'll work out at £1.25 per pack plus £1 for the metal box.

For the rest of this blog post, we'll assume that allstickers are bought in multipacks.

Every album includes Declan Rice (England), Heechan Hwang (South Korea), Alexis Vega (Mexico), Mikel Merino (Spain), Danley Jean Jacques (Haiti) and Nicolas Jackon (Sengal)

How many stickers will I need?

After you've stuck your first 6 stickers in, you need to collect 974 more stickers.Following the same method as I did in 2018, we can work out that if you want to get all of these from packs of random stickers, thenyou should expect to buy 7265 stickers (including the 4 packs in your starter pack). Including the cost of the starter pack, buying all these stickers will cost£1297.35.

But as always, you don't have to spend this much, as Panini let you order the last $n$ stickers (for any value of $n$ that you choose, although I have strong memoriesof $n$ being 50 when we had to send off paper sticker ordering forms when I was at school). On the Panini website, they're selling missing stickersfor 45p per sticker*.

If you decide to order the last 50 stickers, then you should expect to buy 2883 stickers to get the rest of the stickers before ordering the last 50. In total this willcost you £532.81. That's a big saving.

What is the best number of stickers to order?

If you want to minimise the amount that you have to spend to complete this sticker book, you should choose your value of $n$ wisely. This plot shows the expected total costof completing the sticker book if you decide on different numbers of stickers to order:

The lowest expected cost is achieved when ordering 386 stickers: £334.54. That's an even bigger saving, although 386 does feel to me like an unreasonably large value for $n$.

Swapping

As always, the cheapest way to complete the sticker book is to convince more of your friends to collect too so that you can swap your duplicate stickers with them. You could getthe price down as low as £174.99 if you manage to swap every single duplicate that you get.

If you decide to set $n$ to 0, there's a good chance that going out now and buying 200 starter packs for £998 to give to your friends make them start collecting will mean spending less money in totalthan trying to complete your sticker book on your own...

Edit: The first version of this blog post guessed that the missing stickers would cost 36p each, as this was the cost for other recent collections. The numbers have been updated to matchthe 45p that Panini are actually charging. The original lowest cost based on 36p was ordering 483 stickers to give a total cost of £295.78.

A new puzzle every day

20 Apr 2026 12:00:00 GMT

This is an article that I wrote forChalkdust issue 23, and thenew puzzle it introduces appears on the cover of issue 23.

In January, I paid a visit to MathsWorld, the recently opened maths discovery centrein London, alongside some other members of the Chalkdust team.One of the highlights of the trip was playing the two-player game Genius Square.

In Genius Square, you start with a six-by-six board and roll seven dice. These dice tell you whereto place seven cylindrical blocks, for example:

The dice

The board with the cylinders placed and the pieces

The two players then race to fit the pieces shown above into the remaining space on theboard. The pieces that the players have arethe five tetrominoes,the two triominoes,a domino, and a single square (or monomino); these are all the shapes you can make by gluing together up to four squares (if rotations and reflections are considered the sameshape).

If you want to ruin/improve your copy of Chalkdust, you could cut out the pieces shown above andtry to fit them in the board to the left.

There's some clever design in this game:if, instead of rolling the dice, you were to randomly pick any set of seven spaces to place the cylinders,the puzzle is not guaranteed to have a solution.The locations printed on the dice have been carefully chosen so that any combination that you can rollleads to a solvable puzzle.

A puzzle-a-day

The Genius Square puzzle is similar to another rearrangement puzzle: the puzzle-a-day calendar,created by the Norwegian puzzle makers DragonFjord.

The puzzle-a-day board and pieces

In this puzzle, you are given the pieces belowand asked to place them on the board to cover everything excepttoday's date. For example, on 22 July, you could place the pieces like this:

A solution of puzzle-a-day for 22 July

DragonFjord make and sell wooden and plastic versions of puzzle-a-day, whichyou can buy from Maths Gear—whoalso provide the top prize for the crossnumber—toavoid the cost of shipping directly from Norway.

In puzzle-a-day, it's possible to arrange the pieces to make every single combinationof a number and a month, including days that don't exist like 31 September and 30 February.

A solution of puzzle-a-day for 31 September?!

While we were considering options for the cover of this issue, we discussed putting somethinglike Genius Square on the cover, and I began to wonderif it would be possible to makea puzzle like puzzle-a-day but where it was only possible to make days that actually appear on thecalendar.

A new puzzle

After spending a while scribbling on squared paper and getting nowhere,I had an idea: I could put the months in regions that were disconnected from the day numbers. Then,by carefully choosing the shape of the month regions and the arrangement of the dates, I could forcethe solver to use different combinations of pieces on the day numbers for different months.

Once I'd had this idea, I threw together some Python code that could see which day numbersyou could and couldn't leave uncovered with a set of pieces, and waited for it to find a goodset of pieces. It found this board and these pieces:

The board and pieces for the new puzzle

As in Tetris, I've named the pieces after letters that they vaguely resemble.

In January, March, May, July, August, October and December, you have to usea P, an O and the A in the month regions.The remaining pieces can make any day from 1 to 31.

In April, June, September and November, you need to use the C, an O and the A in the month regions.This leaves pieces that can make any day from 1 to 30, but importantly can't make 31.

In February, you need to use both Os and the C in the month regions.This leaves pieces that can make any day from 1 to 29, but not 30 or 31.

Now all we need to do is find another new arrangement that somehow works differently in leap and non-leap years...

Mixing Wordle with other games

06 Apr 2026 12:00:00 GMT

I've recently been thinking about Wordle,the Mastermind-styleword guessing game, that everyone was playing and talking about in 2020. Specifically, I've beenthinking about mixing Wordle with other games.

Tetris

The first game I tried mixing Wordle with is Tetris, the addictive puzzle game that madetetrominos famous.

In Wordle, letters that you guess are coloured green (letter in the right place),yellow (letter that's in the answer but in the wrong place), or grey (letter not in the answer).Inspired by noticing that the yellow squares in one of my Wordle attempts made a Tetris S-block, Icreated a new bot on Bluesky: @wordle-tetris.bsky.social.

This bot posts a set of six words each day that make the next frame in a game of Tetris whenused on Wordle. Every time, the bot is given a new piece, it quickly rotates and moves it intowhat it thinks is the best position (24 hoursbetween frames gives the bot a lot of time to press the buttons to do this) then lets the block fall.Currently, the bot does not move or rotate the block as it falls.For example, the bot's first post was:

L	U	R	I	D
P	U	L	U	T
A	A	R	G	H
R	H	O	N	E
B	R	I	D	E
R	E	F	A	N

The following day it posted:

W	E	B	B	Y
T	R	A	I	N
A	T	R	I	A
L	E	V	E	L
L	I	M	M	A
W	I	P	E	D

It continued until it posted this on the fifth day:

B	U	N	D	U
K	A	P	P	A
M	A	C	C	A
D	U	R	A	L
H	O	K	U	M
M	O	I	L	E

Then a second piece started to fall:

C	O	R	K	Y
L	E	F	T	E
D	O	R	S	A
F	E	D	E	X
S	T	R	O	P
E	T	A	P	E

On the eighth day,the bot reached its first game over, as there was no valid word that could give this row:

After a couple of games, I updated the code that runs the bot so that it now pickscommonly used words whenever it can. This makesthe bot's more recent postslook a lot more like actual guesses someone could have made:

B	R	E	A	D
M	E	T	E	R
K	A	R	M	A
L	I	K	E	D
A	P	A	R	T
T	O	U	G	H

In Wordle, it's impossible to get four green squares and one yellow square in a row, as if fourletters are correct, the other letter cannot be a correct letter in the wrong position. This meansthat Wordle Tetris bot can never clear more than two lines at once, as a line with four greens anda yellow must appear at some point on the way to the clearance. the bot can clear two lines at oncethough if it mangages to set up something like this:

If you prefer your explanations in video form, Ayliean made a funYouTube short about the Wordle Tetris bot.And if you're interested, the code that runs the bot is available on Codeberg.

Pokémon Blue

A Game Boy with Pokémon Wordle inputs

The other game that I've recently mixed Wordle with is Pokémon Blue.Whenever I play Pokémon on an emulator, I useW, A, S and D for the arrow keys,K and J for the A and B buttons,and O and I for the start and select buttons. In February, I started playing a game with thekeyboard input used to play both Pokémon Blue and Wordle.

To give myself a chance ofgetting some way through the game, I'm allowed to know the solution to the day's Wordle and canuse it to play which words to use to get as many useful key presses as I can. Each day's videois posted a day late to avoid spoiling Wordle for anyone still playing it.

Just over a week after starting, I'd pressed J and K enough to getthrough the introduction and started walking around. After another ten days,I picked Bulbasaur to be my starter pokémon thenpromptly started a ten day long battle against my rival.

Probably the best day so far was day 46, when Imanaged to find words to attack and knock out a pidgey, get thorough the post-battle text, thenwalk four steps.The full game so far is on this YouTube playlist.

So far, I've been playing for 53 days and have got about a third of the way along Route 1.The current Pokémon Bluespeedrun record did this in 2 minutes and 51 seconds and went on to complete the game in 1 hour and 43 minutes.Using this as a rough estimate of the proportion of the game completed, it looks like it'll takearound 5 and a quarter years to complete Pokémon Wordle. See you on Victory Road in May2031...

Christmas (2025) is over

08 Feb 2026 12:00:00 GMT

Welcome to 2026 everyone! It's time to reveal the answers to the Advent Calendar puzzles and announce the winners.But first, some good news: with your help, Santa was able to order a new sleigh and Christmas was saved!

Now that the competition is over, the questions and all the answers can be found at mscroggs.co.uk/puzzles/advent2025.Before announcing the winners, I'm going to go through some of my favourite puzzles from the calendar and a couple of other interesting bits and pieces.

Highlights

My first highlight is the puzzle from 9 December. I like this puzzle, and enjoy how it lookslike a complicated counting puzzle at first, but there's a much simpler method available...

9 December

In a 3 by 5 grid of squares, if a line is drawn from the bottom left corner to the top right corner, it will pass through 7 squares:

In a 251 by 272 grid of squares, how many squares will a line drawn from the bottom left corner to the top right corner pass through?

Show answer & extension

My next highlight is the puzzle on 17 December. This is a highlight because I just really like the puzzle.

17 December

A sequence of zeros and ones can be reduced by writing a 0 or 1 under each pair of numbers: 1 is written if the numbers are the same, 0 is written if they are not.This process can be repeated until there is a single number. For example, if we start with the sequence 1, 1, 1, 0, 1 (of length 5), we get:

The final digit is a 1.

How many sequences of zeros and ones of length 10 are there that when reduced lead to the final digit being a 1?

Show answer

My final highlight is the puzzle from 18 December, as I always enjoy a surprise Fibonacci.

18 December

There are 5 different ways to make a set of numbers between 1 and 5 such that the smallest number in the set is equal to the number of numbers in the set. These 5 sets are: {1}, {2, 3}, {2, 4}, {2, 5} and {3, 4, 5}.

How many ways are there to make a set of numbers between 1 and 14 such that the smallest number in the set is equal to the number of numbers in the set?

Show answer

Hardest and easiest puzzles

Once you've entered 24 answers, the calendar checks these and tells you how many are correct. I logged the answers that were sentfor checking and have looked at these to see which puzzles were the most and least commonly incorrect. The bar chart below shows the total numberof incorrect attempts at each question.


1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
Day

It looks like the hardest puzzles were on 21,15 and14 December;and the easiest puzzles were on3,10,5 and8 December.

Ordering the sleigh

To finish the Advent calendar, you were tasked with ordering the correct parts for a new sleigh. The answers to all the puzzles were required tobe certain of which combination of parts was needed, but it was possible to reduce the number of optionsThis graph shows how many people successfully ordered a sleigh on each day:

1	4	21	112	53	31	11	7	10	25	10
21	22	23	24	25	26	27	28	29	30	31
Day

The winners

And finally (and maybe most importantly), on to the winners: 242 people managed to order a slieigh. That's a record high number!

27	43	85	79	133	223	150	192	180	237	242
2015	2016	2017	2018	2019	2020	2021	2022	2023	2024	2025
Year

From the correct answers, the following 10 winners were selected:

Adrienne Hestenes
Blake Segler
Johannes C. Huber
Matthew Harding
Mr J Winfield
Pamela Docherty
Pete Chare
Steve Blay
Tim Dufton
Vinny R

Congratulations! Your prizes will be on their way shortly.

The prizes this year include 2025 Advent calendar T-shirts. If you didn't win one, but would like one of these, I've made them available to buy at merch.mscroggs.co.uk alongside the T-shirts from previous years.

Additionally, well done to100118220919SantaSixSeven9, A Hall, Aaron Kunze, Aisha Arroyo, Alan Buck, Alan Shotkin, alek2ander, Alex Bolton, Alex Slaughter, Allan Taylor, Anastasia P, Anastasiia, Andrea Chlebikova, Andrew Brady, Andrew Fermor, Andrew Roy, Andrew T, Andrew Thomson, Andy Ennaco, Artie Smith, Ashley Jarvis, B Witt, Becky Russell, beeplusdub, Ben Baker, Ben Boxall, Ben J, Ben Reiniger, Ben Semanko, Ben Tozer, Ben Weiss, Bill, Bill Russ, Bob B, Brian Wellington, Carmen, Cathy Hooper, Chad Smith, Chloe, Chris Dettmar, Chris Eagle, Chris Hazell, Chris Hellings, Christopher Adams, Clint Cabrera, Colin Beveridge , Colin Brockley, Connie, Cool Beans, Corbin Groothuis, Cristian Sbârciog, dajedrek, Dan Colestock, Dan May, Dan Whitman, Daniel, Daniel Low, Danny, Dave, Dave Budd, David, David Ault, David Berardo, David Carey, David Fox, David Kendel, Deborah Tayler, Dhruv Pisharody, Donagh, Donald Anderson, Dylan, Dylan Madisetti, Echo231, Elijah Kuhn, Elizabeth Madisetti, Ellie Winters, Elytre, Emily Troyer, Eoin Davey, epsilon, Eric Kolbusz, Erick Lee, Evan and Dana, Evan Denney, Evan S, Ewan Beetham, F Z, Fabien Friedli, Fionn Woodcock, Frank Kasell, Fred Verheul, Gabriella , Gareth McCaughan, Gary M. Gerken, Gary Male, Gert-Jan, Gregory Wheeler, Guillermo Heras Prieto, H.Hung, Hannah Harris, Heerpal, Helen, Herschel, Holly Carnes, Håkon Balteskard, IanAllenBird, Iris Lasthofer, Isabel Turner, Ivan Andrus, Jacob, Jamas Enright, James Chapman, James Dolengewicz, James Swenson, Jan Zemba, Jay Winter, JDev, Jean-Noël Monette, Jen Sparks, Jessica Marsh, Joe Gage, Johan, Jon Palin, Jonathan Chaffer, Jonathan Thiele, Jorge del Castillo Tierz, Josh Hernandez, Judith W, Kat, Kat Yates, Katerina Stergiopoulou, Katie Steckles, Keerthana, Kevin Docherty, Kirsty Fish, Kristen Koenigs, Larry Goddard, lastrun, Lazar Ilic, LDufton, Leif Cooper, Lewis Dyer, Lisa Stambaugh, Lise Andreasen, Lizzie McLean, Lorelei, Louis, Lucas Bowman, Magnus Eklund, Mair A-W, Marc G., Marco van der Park, Mark Lydon, Mark Stambaugh, Martin Harris, María Jesús Rapanague, mathmandan, Matthew Courtney, Matthew Schulz, Max, Maya, Michael, Mihai Zsisku, Mike Graczyk, Millie, Mimi Manning, Mister Ron, Monopoler, Nazneen Molu, Neil Bastian, Nick C, Nick Keith, Nikos I., Noah Molder, Noah O, Oli M, Olov, Pablo Carballeira, Peter Krol, Peter Rowlett, Pierce R, Pollyanna, Pythialouise, Rachel Sheridan, RADina, Rashi, Ray Arndorfer, Richard O, Rob Reynolds, Robert, Robert Allwright, Roger, Roni, Ronno, Rosamund, Rosie Paterson, Sadie Robinson, Sam Dreilinger, Sam Peterson, Sam123guy, Samuel Wilson , Scio Durango, Scott, Sean Henderson, Seth Cohen, Shannon Stranahan, Shivanshi, Simon English, sjlxndr, stephen kirkham, Stephen Royle, Tamas Toth, Tarka Burrell, Tehnuka, The Connors of York, Thomas O'Neill, Tim B, Timothy Conlan, Tina Furer, Tino, Tony Mann, Travis, tripleboleo, UsrBinPRL, Valentin VĂLCIU, Victor MIller, Vinayak, Will Bayliff, Willem, Yasha, and zookwho all also completed the Advent calendar but were too unlucky to win prizes this time or chose to not enter the prize draw.

See you all next December, when the Advent calendar will return.

Advent calendar 2025

31 Dec 2025 12:00:00 GMT

25 December

It's nearly Christmas and something terrible has happened: after years of gradual wear and tear, Santa's magic sleigh has fallen apart.You need to help Santa build a new sleigh so that he can deliver presents before Christmas is ruined for everyone.

The magic sleigh was built from parts bought from magic factories all over the world, and only the exact combination of parts will be magic enough to let Santa deliver allthe world's presents on Christmas Eve—but it's been a long time since a magic sleigh was last built and no-one can remember exactly which combination of parts is needed.You need to use the clues below to work out which parts to order to build a new magic sleigh. There are five parts that you need to order(runners, a chassis, a present holder, a seat for Santa, and a front of the sleigh with reins). There are nine suppliers (numbered from 1 to 9) that Santa can order parts from.

The chassis should not be ordered from 9

The chassis should not be ordered from 7

One of the parts should be ordered from 2

The chassis should not be ordered from 4

The front should not be ordered from 5

The chassis should not be ordered from 6

Santa's seat should be ordered from 3, 5, or 4

The front should not be ordered from 8

One of the parts should be ordered from 4

The runners should not be ordered from a factor of 175

Today's clue and the clue on 19 December are false

The chassis should not be ordered from 5

The chassis should not be ordered from 1

The chassis should not be ordered from 3

Santa's seat should be ordered from 6, 7, or 5

The runners should be ordered from 5

The chassis should not be ordered from 4

One of the parts should be ordered from 8

The front should not be ordered from 1

Exactly 4 of the clues are false

The chassis should not be ordered from 1, 2, or 8

The chassis should be ordered from 4

One of the parts should be ordered from 1

The chassis should not be ordered from 9

The chassis should not be ordered from 7

One of the parts should be ordered from 2

The chassis should not be ordered from 4

The front should not be ordered from 5

The chassis should not be ordered from 6

Santa's seat should be ordered from 3, 5, or 4

The front should not be ordered from 8

One of the parts should be ordered from 4

The runners should not be ordered from a factor of 175

Today's clue and the clue on 19 December are false

The chassis should not be ordered from 5

The chassis should not be ordered from 1

The chassis should not be ordered from 3

Santa's seat should be ordered from 6, 7, or 5

The runners should be ordered from 5

The chassis should not be ordered from 4

One of the parts should be ordered from 8

The front should not be ordered from 1

Exactly 4 of the clues are false

The chassis should not be ordered from 1, 2, or 8

The chassis should be ordered from 4

One of the parts should be ordered from 1

You can use this page to try ordering parts for a sleigh.

24 December

3 and 5 are both factors of 2025, and 3 and 5 are the only two prime numbers that are factors of 2025.

What is the largest three-digit number that has both 3 and 5 as factors and no other prime numbers as factors?

23 December

153 is equal to the sum of the cubes of its digits: 1³ + 5³ + 3³.

There are three other three-digit numbers that are equal to the sum of the cubes of their digits. What is the largest of these numbers?

22 December

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10.Today's number is the product of the numbers in the red boxes.

= 11

= 1

–

= 1

=
1

=
0

=
1

21 December

There are ten ways to make a list of four As and Bs that don't contain an even* number of consecutive As:

B,B,B,B
A,B,B,B
B,A,B,B
B,B,A,B
B,B,B,A

A,B,A,B
A,B,B,A
B,A,B,A
A,A,A,B
B,A,A,A

How many ways are there to make a list of eleven As and Bs that don't contain an even number of consecutive As?

* We don't count 0 consecutive As as being an even number of consecutive As.

20 December

A number is called a perfect power if it is equal to n^k for some integer n and some integer k > 1. 2025 is a perfect power (45²)and 23 more than 2025 is also a perfect power (2¹¹).

What is the only three-digit perfect power that is 29 less than another perfect power?

19 December

Eve uses the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 to write five square numbers (using each digit exactly once). What is largest square number that she made?

18 December

How many ways are there to make a set of numbers between 1 and 14 such that the smallest number in the set is equal to the number of numbers in the set?

17 December

The final digit is a 1.

How many sequences of zeros and ones of length 10 are there that when reduced lead to the final digit being a 1?

16 December

–

= 1

–

= 37

= 17

=
2

=
1

=
2

15 December

The odd factors of 2025 are 1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675 and 2025. There are 15 of these factors and 15 is itself an odd factor of 2025.

What is the smallest three-digit number whose number of odd factors is itself an odd factor of the number?

14 December

There are five ways to make a list of four As and Bs that don't contain an odd number of consecutive As:

B,B,B,B
A,A,B,B
B,A,A,B
B,B,A,A
A,A,A,A

How many ways are there to make a list of eleven As and Bs that don't contain an odd number of consecutive As?

13 December

Today's number is given in this crossnumber. No number in the completed grid starts with 0.

1	2	3
4
	5

Across
1	Today's number.	(3)
4	Two times today's number.	(3)
5	The product of the digits of today's number.	(2)

Down
1	The sum of the digits of today's number.	(2)
2	Two more than an anagram of today's number.	(3)
3	A multiple of 101.	(3)

12 December

Mary uses the digits 1, 2, 3, 4, 5, 6 and 7 to make two three-digit numbers and a one-digit number (using each digit exactly once). The sum of her three numbers is 1000.

What is the smallest that the larger of her two three-digit numbers could be?

11 December

Holly added up 3 consecutive numbers starting at 10, then added up the next 3 consective numbers, then found the difference between her two totals:

10 + 11 + 12 = 33
13 + 14 + 15 = 42
42 – 33 = 9

Ivy added up n consecutive numbers starting at m, then added up the next n consecutive numbers, then found the difference between her two totals.The difference was 203401. What is the largest possible value of n that Ivy could have used?

10 December

2025 is the smallest number with exactly 15 odd factors.

What is the smallest number with exactly 16 odd factors?

9 December

In a 3 by 5 grid of squares, if a line is drawn from the bottom left corner to the top right corner, it will pass through 7 squares:

In a 251 by 272 grid of squares, how many squares will a line drawn from the bottom left corner to the top right corner pass through?

8 December

Angel wrote out a muliplication square for the numbers from 1 to 3 (the table has the numbers 1 to 3 in the top row and left column, then every otherentry is equal to the number at the top of its column multiplied by the number at the left of its row):

1	2	3
2	4	6
3	6	9

The sum of the numbers in the bottom row is 18. The sum of all the numbers in the table is 36.

Angel then wrote out another multiplication square with the numbers from 1 to $n$. The sum of all the numbers in the new table is 2025. What is thesum of the numbers in the bottom row of the new table?

7 December

Carol organised a knockout competition in December 2024, which 6 people entered. There were 2 matches in the first round with the remaining two players given byes(so they went into the next round without playing a match). The second round was made up of two semi-finals, then one final match was played to decide the winner.In total 5 matches were played.

This year, Carol is organising the competition again, but it has become a lot more popular: 355 people have entered.While planning the tournament, she can decide which rounds to give people byes in.What is the smallest number of matches that could be included in the tournament?

6 December

= 20

–

= 26

–

= 28

=
32

=
2

=
11

5 December

The number 36 is equal to two times the product of its digits.

What is the only (strictly positive) number that is equal to four times the product of its digits?

4 December

Some numbers can be written as the sum of four consecutive numbers, for example: 142 = 34 + 35 + 36 + 37.

What is the mean of all the three-digit numbers that can be written as the sum of four consecutive numbers?

3 December

Holly picks the number 513, reverses it to get 315, then adds the two together to make 828.

Ivy picks a three-digit number, reverses it, then adds the two together to make 968. What is the smallest number that Ivy could have started with?

2 December

Eve writes down the numbers from 1 to 10 (inclusive). In total she write down 11 digits.

Noel writes down the number from 1 to 100 (inclusive). How many digits does he write down?

1 December

Some numbers contain a digit more than once (eg 313, 111, and 144). Other numbers have digits that are all different (eg 123, 307, and 149).

How many three-digit numbers are there whose digits are all different?

Christmas card 2025

04 Dec 2025 12:00:00 GMT

As usual, I spent some time this November, designing this year's Chalkdust puzzle Christmas card(with help from TD and Jacob).

The card contains 12 puzzles: 8 in the green section, and 4 in the red or yellow section. By colouring the two squares on the front of the card containing every pair of digits in each answer(eg if an answer in the green section were 3305, you would colour the squares containing 33, 30 and 05 green), you will reveal a Christmasthemed picture.

If you're in the UK and want some copies of the card to send to your maths-loving friends, you can order them from my Ko-Fi shop.

If you want to try the card yourself, you can download this printable A4 pdf. Alternatively, you can find the puzzles below and type the answers in the boxes. The answers will automatically be used to colour in the appropriate squares.

Green
1.	What is the sum of all the odd integers between 0 and 12?	Answer
2.	What is the sum of all the odd integers between 0 and 44444?	Answer
3.	Carol has more than one bauble. When she shares them equally between 3 people, there is one bauble left over. When she shares them equally between 4 people, there is still one bauble left over. What is the smallest number of baubles that Carol could have?	Answer
4.	Carol has more than one bauble. When she shares them equally between 2, 3, 4, 5, 6, 7, 8, or 9 people, there is one bauble left over. What is the smallest number of baubles that Carol could have?	Answer
5.	What is the smallest three-digit positive integer whose digits are all non-zero and different?	Answer
6.	How many positive integers are there whose digits are all non-zero and different?	Answer
7.	In a Christmas game, you can win either 4 points or 5 points on your turn, and the game can last any number of turns. What is the largest number of points that it is impossible to end the game on?	Answer
8.	In a different Christmas game, you can win either 495371 points or 2921695 points on your turn, and the game can last any number of turns. What is the largest number of points that it is impossible to end the game on?	Answer
Red or yellow
9.	What is the smallest two-digit positive integer that cannot be written as the sum of consecutive integers?	Answer
10.	What is the smallest four-digit positive integer that can be written as the sum of exactly four consecutive integers?	Answer
11.	Holly drew 18 points on the circumference of a circle then drew straight lines connecting each pair of points. How many straight lines did she draw?	Answer
12.	Holly drew some points on the circumference of a circle then drew straight lines connecting each pair of points. She drew 166047976 straight lines. How many points did she draw?	Answer

Advent calendar 2024

31 Dec 2024 12:00:00 GMT

25 December

It's nearly Christmas and something terrible has happened: there's been a major malfunction in multiple machines in Santa's toy factory, andnot enough presents have been made. Santa has a backup warehouse full of wrapped presents that can be used in the case of severe emergency, but the warehouse is locked.You need to help Santa work out the code to unlock the warehouse so that he can deliver the presents before Christmas is ruined for everyone.

The information needed to work out the code to the warehouse is known by Santa and his three most trusted elves: Santa is remembering a three-digit number,and each elf is remembering a one-digit and a three-digit number. If Santa and the elves all agree that the emergency warehouse should be opened, they can work out the code for the door as follows:

Santa tells his three-digit number to the first elf.
The first elf subtracts her three-digit number then multiplies by her one-digit number. She tells her result to the second elf.
The second elf subtracts his three-digit number then multiplies by his one-digit number. He tells his result to the third elf.
The third elf subtracts their three-digit number then multiplies by their one-digit number. Their result is a five-digit number that is the code to unlock the warehouse.

But this year, there is a complication: the three elves are on a diplomatic mission to Mars to visit Martian Santa and cannot be contacted, so you need to piece together theirnumbers from the clues they have left behind:

138 is an elf's 3-digit number.

The third elf's 1-digit number is not 2.

The first elf's 1-digit number is not 6.

144 is an elf's 3-digit number.

The third elf's 1-digit number is not 7.

The HCF of the first and third elves' 1-digit numbers is not 1, 2, or 8.

The second elf's 1-digit number is not 2, 8, or 1.

The third elf's 1-digit number is not a factor of 575.

The first elf's 1-digit number is not 9.

The third elf's 1-digit number is not 4.

The third elf's 1-digit number is not 1.

The third elf's 1-digit number is not 9.

990 is a multiple of an elf's 3-digit number.

The first elf's 1-digit number is not 7.

The HCF of 851 and the third elf's 3-digit number is 1.

The second elf's 1-digit number is not 6, 2, or 5.

Santa's number is 444.

The first elf's 1-digit number is not 5.

The second elf's 1-digit number is not 1, 7, or 9.

The second elf's 1-digit number is not 4, 9, or 5.

One of the digits of the second elf's 3-digit number is 9.

The HCF of 256 and the second elf's 3-digit number is 1.

The first elf's 1-digit number is not a factor of 202.

The first elf's 1-digit number is not 3.

138 is an elf's 3-digit number.

The third elf's 1-digit number is not 2.

The first elf's 1-digit number is not 6.

144 is an elf's 3-digit number.

The third elf's 1-digit number is not 7.

The HCF of the first and third elves' 1-digit numbers is not 1, 2, or 8.

The second elf's 1-digit number is not 2, 8, or 1.

The third elf's 1-digit number is not a factor of 575.

The first elf's 1-digit number is not 9.

The third elf's 1-digit number is not 4.

The third elf's 1-digit number is not 1.

The third elf's 1-digit number is not 9.

990 is a multiple of an elf's 3-digit number.

The first elf's 1-digit number is not 7.

The HCF of 851 and the third elf's 3-digit number is 1.

The second elf's 1-digit number is not 6, 2, or 5.

Santa's number is 444.

The first elf's 1-digit number is not 5.

The second elf's 1-digit number is not 1, 7, or 9.

The second elf's 1-digit number is not 4, 9, or 5.

One of the digits of the second elf's 3-digit number is 9.

The HCF of 256 and the second elf's 3-digit number is 1.

The first elf's 1-digit number is not a factor of 202.

The first elf's 1-digit number is not 3.

You can to open the door here.

24 December

There are 343 three-digit numbers whose digits are all 1, 2, 3, 4, 5, 6, or 7. What is themean of all these numbers?

23 December

In a grid of squares, each square is friendly with itself and friendly with every square that is horizontally, vertically, or diagonally adjacent to it (and is not friendly with any other squares).In a 5×5 grid, it is possible to colour 8 squares so that every square is friendly with at least two coloured squares:

It it not possible to do this by colouring fewer than 8 squares.

What is the fewest number of squares that need to be coloured in a23×23 grid so that every square is friendly with at least two coloured squares?

22 December

= 3

–

= 6

–

= 16

=
1

=
13

=
16

21 December

Noel wants to write a different non-zero digit in each of the five boxes below so thatthe products of the digits of the three-digit numbers reading across and down are the same.

What is the smallest three-digit number that Noel could write in the boxes going across?

20 December

p(x) is a polynomial with integer coefficients such that:

p(0) > 0;
if x is a real number,4x – 9 < p(x) < x² – 2x + 2.

What is p(23)?

19 December

There are 9 integers below 100 whose digits are all non-zero and add up to 9:9, 18, 27, 36, 45, 54, 63, 72, and 81.

How many positive integers are there whose digits are all non-zero and add up to 9?

18 December

If k = 21, then 28k ÷ (28 + k) is an integer.

What is the largest integer k such that 28k ÷ (28 + k) is an integer?

17 December

The number 40 has 8 factors: 1, 2, 4, 5, 8, 10, 20, and 40.

How many factors does the number 2²⁶×5×7⁵×11² have?

16 December

= 46

= 1

–

= 1

=
1

=
12

=
45

15 December

The number 2268 is equal to the product of a square number (whose last digit is not 0) and the same square number with its digits reversed:36×63.

What is the smallest three-digit number that is equal to the product of a square number (whose last digit is not 0) and the same square number with its digits reversed?

14 December

15³ is 3375. The last 3 digits of 15³ are 375.

What are the last 3 digits of 15^1234567890?

13 December

Today's number is given in this crossnumber. No number in the completed grid starts with 0.

1	2	3
4
5

Across
1	Today's number.	(3)
4	Two times 5A.	(3)
5	A multiple of 1.	(3)

Down
1	Sum of digits is 15.	(3)
2	Sum of digits is 19.	(3)
3	Three times 5A.	(3)

12 December

Holly picks a three-digit number. She then makes a two-digit number by removing one of the digits.The sum of her two numbers is 309. What was Holly's original three-digit number?

11 December

There are 6 sets of integers between 1 and 5 (inclusive) that contain an odd number of numbers whose median value is 3:

{3}
{1,3,4}
{2,3,4}
{1,3,5}
{2,3,5}
{1,2,3,4,5}

How many sets of integers between 1 and 11 (inclusive) are there that contain an odd number of numbers whose median value is 5?

10 December

A number is a palindrome if it's the same when its digits are written in reverse order.

What is the sum of all the numbers between 10 and 100 that are palindromes?

9 December

–

= 8

–

= 9

=
12

=
1

=
3

8 December

It is possible to arrange 4 points on a plane and draw non-intersecting linesbetween them to form 3 non-overlapping triangles:

It is not possible to make more than 3 triangles with 4 points.

What is the maximum number of non-overlapping triangles that can be made by arranging 290 points on a planeand drawing non-intersecting lines between them?

7 December

What is the obtuse angle in degrees between the minute and hour hands of a clock at 08:22?

6 December

The number n has 55 digits. All of its digits are 9.What is the sum of the digits of n³?

5 December

The sum of 11 consecutive integers is 2024. What is the smallest of the 11 integers?

4 December

The geometric mean of a set of n numbers is computed by mulitplying all thenumbers together, then taking the nth root.

The factors of 9 are 1, 3, and 9. The geometric mean of these factors is

$$\sqrt[3]{1\times3\times9}=\sqrt[3]{27}=3$$

What is the smallest number where the geometric mean of its factors is 13?

3 December

There are 5 ways to write 5 as the sum of positive odd numbers:

1 + 1 + 1 + 1 + 1
1 + 1 + 3
3 + 1 + 1
1 + 3 + 1
5

How many ways are there to write 14 as the sum of positive odd numbers?

2 December

14 is the smallest even number that cannot be obtained by rolling two 6-sided dice and findingthe product of the numbers rolled.

What is the smallest even number that cannot be obtained by rolling one hundred 100-sided diceand finding the product of the numbers rolled?

1 December

Eve writes down five different positive integers. The sum of her integers is 16. What isthe product of her integers?

Advent calendar 2023

31 Dec 2023 12:00:00 GMT

25 December

It's nearly Christmas and something terrible has happened: a machine in Santa's toy factory has malfunctioned, and is unable to finish building allthe presents that Santa needs.You need to help Santa work out how to fix the broken machine so that he can build the presents and deliver them before Christmas is ruined for everyone.

Inside the broken machine, there were five toy production units (TPUs) installed at sockets labelled A to E. During the malfunction, these TPUs wereso heavily damaged that Santa is unable to identify which TPU they were when trying to fix the machine. The company that supplies TPUs builds 10 different units, numbered from 0 to 9.You need to work out which of the 10 TPUs needs to be installed in each of the machine's sockets, so that Santa can fix the machine. It may be that two or more of the TPUs are the same.

You can attempt to fix the machine here.

24 December

When written in binary, the number 235 is 11101011. This binary representation starts andends with 1 and does not contain two 0s in a row.

What is the smallest three-digit number whose binary representation starts andends with 1 and does not contain two 0s in a row?

23 December

There are 18 ways to split a 3 by 3 square into 3 rectangles whose sides all have integer length:

How many ways are there to split a 10 by 10 square into 3 rectangles whose sides all have integer length?

22 December

There are 4 ways to pick three vertices of a regular quadrilateral so that they form a right-angled triangle:

In another regular polygon with $n$ sides, there are 14620 ways to pick three vertices so that they form a right-angled triangle. What is $n$?

21 December

There are 6 two-digit numbers whose digits are all 1, 2, or 3 and whose second digit onwardsare all less than or equal to the previous digit:

How many 20-digit numbers are there whose digits are all 1, 2, or 3 and whose second digit onwardsare all less than or equal to the previous digit?

20 December

There are 6 different ways that three balls labelled 1 to 3 can be put into two boxes labelledA and B so that no box is empty:

How many ways can five balls labelled 1 to 5 be put into four boxes labelled A to D so thatno box is empty?

19 December

–

= 7

–

= 0

–

= 2

=
4

=
35

=
18

18 December

Some numbers can be written as the product of two or more consecutive integers, for example:

$$6=2\times3$$$$840=4\times5\times6\times7$$

What is the smallest three-digit number that can be written as the product of two or more consecutive integers?

17 December

If you expand $(a+b+c)^2$, you get $a^2+b^2+c^2+2ab+2ac+2bc$.This has 6 terms.

How many terms does the expansion of $(a+b+c+d+e+f)^5$ have?

16 December

Some numbers can be written as the sum of two or more consecutive positive integers, for example:

$$7=3+4$$$$18=5+6+7$$

Some numbers (for example 4) cannot be written as the sum of two or more consecutive positive integers.What is the smallest three-digit number that cannot be written as the sum of two or more consecutive positive integers?

15 December

The arithmetic mean of a set of $n$ numbers is computed by adding up all the numbers, thendividing the result by $n$.The geometric mean of a set of $n$ numbers is computed by multiplying all the numbers together, thentaking the $n$th root of the result.

The arithmetic mean of the digits of the number 132 is $\tfrac13(1+3+2)=2$.The geometric mean of the digits of the number 139 is $\sqrt[3]{1\times3\times9}$=3.

What is the smallest three-digit number whose first digit is 4 and for which the arithmetic and geometric means of its digits are both non-zero integers?

14 December

The function $f(x)=ax+b$ (where $a$ and $b$ are real constants) satisfies

$$-x^3+2x^2+6x-9\leqslant f(x)\leqslant x^2-2x+3$$

whenever $0\leqslant x\leqslant3$. What is $f(200)$?

13 December

Today's number is given in this crossnumber. No number in the completed grid starts with 0.

1	2	3
4
5

Across
1	Today's number.	(3)
4	A multiple of 5 times the sum of the digits of 2D.	(3)
5	Greater than two times 3D.	(3)

Down
1	The sum of 5A and 3D.	(3)
2	Nine times the sum of the digits of 4A.	(3)
3	Greater than 2D.	(3)

12 December

What is the smallest value of $n$ such that

$$\frac{500!\times499!\times498!\times\dots\times1!}{n!}$$

is a square number?

11 December

= 15

–

= 10

–

= 3

=
16

=
1

=
30

10 December

How many integers are there between 100 and 1000 whose digits add up to an even number?

9 December

The diagram below shows a rectangle. Two of its sides have been coloured blue. A red line has been drawn from two of its vertices to the midpoint of a side.

The total length of the blue lines is 50cm. The total length of the red lines is also 50cm. What is the area of the rectangle (in cm²)?

8 December

Noel writes the numbers 1 to 17 in a row. Underneath, he writes the same list without the first and last numbers, then continues this until he writes a row containing just one number:

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
		3	4	5	6	7	8	9	10	11	12	13	14	15
etc.

What is the sum of all the numbers that Noel has written?

7 December

There are 8 sets (including the empty set) that contain numbers from 1 to 4 that don't include any consecutive integers:

$\{\}$, $\{1\}$, $\{2\}$, $\{3\}$, $\{4\}$, $\{1,3\}$, $\{1,4\}$, $\{2, 4\}$

How many sets (including the empty set) are there that contain numbers from 1 to 14 that don't include any consecutive integers?

6 December

There are 5 ways to tile a 4×2 rectangle with 2×1 pieces:

How many ways are there to tile a 12×2 rectangle with 2×1 pieces?

5 December

= 15

–

= 15

=
15

4 December

If $n$ is 1, 2, 4, or 6 then $(n!-3)/(n-3)$ is an integer. The largest of these numbers is 6.

What is the largest possible value of $n$ for which $(n!-123)/(n-123)$ is an integer?

3 December

190 is the smallest multiple of 10 whose digits add up to 10.

What is the smallest multiple of 15 whose digits add up to 15?

2 December

Holly adds up the first six even numbers, then adds on half of the next even number. Her total is 49.

Next, Holly adds up the first $n$ even numbers then adds on half of the next even number. This time, her total is 465124. What is $n$?

1 December

Each interior angle of a regular triangle is 60°.

Each interior angle of a different regular polygon is 178°. How many sides does this polygon have?

Advent calendar 2022

31 Dec 2022 12:00:00 GMT

Advent 2022 logic puzzle

It's nearly Christmas and something terrible has happened: an evil Christmas-hater has set three drones loose above Santa's stables. As long as the drones are flying around, Santa is unable totake off to deliver presents to children all over the world.You need to help Santa by destroying the drones so that he can deliver presents before Christmas is ruined for everyone.

Each of the three drones was programmed with four integers between 1 and 20 (inclusive): the first two of these are the drone's starting position; the last two give the drone's daily speed.The drones have divided the sky above Santa's stables into a 20 by 20 grid. On 1 December, the drones will be at their starting position.Each day, every drone will add the first number in their daily speed to their horizontal position, and the secondnumber to their vertical position. If the drone's position in either direction becomes greater than 20, the drone will subtract 20 from their position in that direction.Midnight in Santa's special Advent timezone is at 5am GMT, and so the day will change and the drones will all move at 5am GMT.For example, if a drone's starting position was (1, 12) and its movement was (5, 7), then:

on day 1, it would be at (1, 12);
on day 2, it would be at (6, 19);
on day 3, it would be at (11, 6);
on day 4, it would be at (16, 13);
on day 5, it would be at (1, 20);
on day 6, it would be at (6, 7);
and so on.

You need to calculate each drone's starting position and daily speed, then work out where the drone currently is so you can shoot it down.

Blue's horizontal speed is not 3, 6, or 9.

On 4 December, blue's vertical position was 14.

Blue's vertical speed is not divisible by 2.

On 5 December, orange's vertical position was not 6.

Red's horizontal speed is a factor of 840.

Red's horizontal speed is not a factor of 128.

Blue's vertical speed is not 3, 1, or 5.

Red's vertical speed is 1.

Orange's horizontal speed is 1, 4, or 3.

The highest common factor of blue's horizontal speed and 28 is 1.

On 15 December, the horizontal position of one drone was 7.

Blue's vertical speed is a factor of 840.

Red's horizontal speed is 9, 8, or 7.

On 2 December, red's horizontal position was 5

Orange's horizontal speed is 3, 2, or 4.

Blue's horizontal speed is not a factor of 323.

On 8 December, the vertical position of two drones was 20.

On 16 December, the horizontal position of one drone was 8.

On 14 December, the horizontal position of one drone was 4.

Orange's horizontal speed is 2, 3, or 1.

On 6 December, the vertical position of two drones was 4.

Blue's horizontal speed is not 4, 2, or 5.

Blue's horizontal speed is 1 more or 1 less than a multiple of 5.

Blue's vertical speed is not 4, 7, or 6.

You can attempt to shoot down the drones here.

24 December

The expression $(3x-1)^2$ can be expanded to give $9x^2-6x+1$. Thesum of the coefficients in this expansion is $9-6+1=4$.

What is the sum of the coefficients in the expansion of $(3x-1)^7$?

23 December

How many numbers are there between 100 and 1000 that contain no 0, 1, 2, 3, or 4?

22 December

Ivy makes a sequence by starting with the number 35, then repeatedly making the next term by reversing the digits of the current number and adding 6.The first few terms of this sequence are:

$$35$$$$53+6 = 59$$$$95+6 = 101$$

What is the first number in Ivy's sequence that is smaller than the previous term?

21 December

In the annual tournament of Christmas puzzles, each player must play one puzzle matchagainst each other player. Last year there were four entrants into the tournament (A, B, C, and D),and so 6 matches were played: A vs B, C vs D, A vs D, A vs C, D vs B, and finally B vs C.

This year, the tournament has grown in popularity and 22 players have entered. How manymatches will be played this year?

20 December

The diagram to the right shows (two copies of) quadrilateral ABCD.

The sum of the angles ABC and BCD (green and blue in quadrilateral on the left) is 180°.The sum of the angles ABC and DAB (green and orange in quadrilateral on the left) is also 180°.In the diagram on the right, a point inside the quadrilateral has been used to draw two triangles.

The area of the quadrilateral is 850. What is the smallest that the total area of the two trianglescould be?

19 December

120 is the smallest number with exactly 16 factors (including 1 and 120 itself).

What is the second smallest number with exactly 16 factors (including 1 and the number itself)?

18 December

Noel writes the integers from 1 to 1000 in a large triangle like this:

			1
		2	3	4
	5	6	7	8	9
10	11	12	13	...

The number 12 is directly below the number 6. Which number is directly below the number 133?

17 December

= 10

= 12

–

= 23

=
10

=
12

=
23

16 December

Noel writes the integers from 1 to 1000 in a large triangle like this:

			1
		2	3	4
	5	6	7	8	9
10	11	12	13	...

The rightmost number in the row containing the number 6 is 9.What is the rightmost number in the row containing the number 300?

15 December

There are 3 even numbers between 3 and 9.

What is the only odd number $n$ such that there are $n$ even numbersbetween $n$ and 729?

14 December

Holly draws a line of connected regular pentagons like this:

She continues the pattern until she has drawn 204 pentagons. The perimeter of each pentagon is 5.What is the perimeter of her line of pentagons?

13 December

Today's number is given in this crossnumber. The across clues are given as normal, but the down clues are given in a random order: you must work outwhich clue goes with each down entry and solve the crossnumber to find today's number.No number in the completed grid starts with 0.

1	2	3
4
5

Across
1	A cube number.	(3)
4	A square number.	(3)
5	A multiple of 13.	(3)

Down (in a random order)
Two times 1A. A square number. Today's number.

12 December

The determinant of the 2 by 2 matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is $ad-bc$.

If a 2 by 2 matrix's entries are all in the set $\{1, 2, 3\}$, the largestpossible deteminant of this matrix is 8.

What is the largest possible determinant of a 2 by 2 matrix whose entries are all in the set$\{1, 2, 3, ..., 12\}$?

11 December

There are five 3-digit numbers whose digits are all either 1 or 2 and who do not containtwo 2s in a row: 111, 112, 121, 211, and 212.

How many 14-digit numbers are there whose digits are all either 1 or 2 and who do not containtwo 2s in a row?

10 December

A line is tangent to a curve if the line touches the curve at exactly one point.

The line $y=-160\,000$ is tangent to the parabola $y=x^2-ax$. What is $a$?

9 December

= 20

–

= 0

–

= 12

=
22

=
6

=
2

8 December

The equation $x^5 - 7x^4 - 27x^3 + 175x^2 + 218x = 840$ has five real solutions. What is the product of all these solutions?

7 December

What is the area of the largest triangle that fits inside a regular hexagon with area 952?

6 December

There are 21 three-digit integers whose digits are all non-zero and whose digits add up to 8.

How many positive integers are there whose digits are all non-zero and whose digits add up to 8?

5 December

= 15

= 14

–

= 27

=
9

=
5

4 December

The last three digits of $5^5$ are 125.

What are the last three digits of $5^{2,022,000,000}$?

3 December

Write the numbers 1 to 81 in a grid like this:

$$\begin{array}{cccc}1&2&3&\cdots&9\\10&11&12&\cdots&18\\19&20&21&\cdots&27\\\vdots&\vdots&\vdots&\ddots&\vdots\\73&74&75&\cdots&81\end{array}$$

Pick 9 numbers so that you have exactly one number in each row and one number in each column, and find their sum.What is the largest value you can get?

2 December

What is the smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 7, and 8?

1 December

One of the vertices of a rectangle is at the point $(9, 0)$. The $x$-axis and $y$-axis are both lines of symmetry of the rectangle.

What is the area of the rectangle?

Advent calendar 2021

31 Dec 2021 12:00:00 GMT

Advent 2021 logic puzzle

It's nearly Christmas and something terrible has happened: a saboteur has infiltrated the stables where Santa's reindeer are kept, and has caused all three of Santa's test flights to be unsuccessful.You need to help Santa have a successful test flight so that he can deliver presents before Christmas is ruined for everyone.

In order to have enough magical power to fly with the sleigh, all nine of Santa's reindeer must be fed their favourite food. The saboteur gave one or more reindeer the wrong food before each ofthe three test flights, causing the reindeer to be unable to take off.

In each clue, "before test flight n" means "immediately before test flight n". Before each test flight, each reindeer was fed exactly onefood, and two or more reindeer may have been fed the same food. Two or more reindeer may have the same favourite food. You must use these clues to work out what each reindeer's favourite food is, then complete a test flight by feeding each reindeer the correct food.

Before test flight 2, reindeer 9 was given food 5.

Before test flight 2, reindeer 8 was given food 2

Before test flight 1, reindeer 2 was given food 4.

Before test flight 1, 2 reindeer were given the wrong food.

Before test flight 1, reindeer 9 was given food 6

Before test flight 3, reindeer 9 was given food 1

Before test flight 3, reindeer 5 was not given food 7

Before test flight 3, reindeer 7 was given food that is a factor of 148

Before test flight 2, reindeer 2 was given food 4.

Before test flight 3, reindeer 2 was given food 6.

Reindeer 4's favourite food is a factor of 607

Before test flight 2, reindeer 4 was not given food 9

Before test flight 3, 3 reindeer had the food equal to their number

Before test flight 3, reindeer 7 was not given food 1

Before test flight 3, no reindeer was given food 2

Before test flight 3, 4 reindeer were given the wrong food.

Reindeer 4 was given the same food before all three test flights.

Before test flight 2, 2 reindeer were given the wrong food

Before test flight 2, all the reindeer were given different foods

Before test flight 1, reindeer 7 was not given food 7

Before test flight 1, reindeer 7 was not given food 9

Reindeer 2's favourite food is 4

Before test flight 1, reindeer 8 was given food 3.

Reindeer 1 was given food 1 before all three test flights

You can attempt a test flight here.

24 December

The digital product of a number is computed by multiplying together all of its digits.For example, the digital product of 1522 is 20.

How many 12-digit numbers are there whose digital product is 20?

23 December

I draw the parabola $y=x^2$ and mark points on the parabola at $x=17$ and $x=-6$.I then draw a straight line connecting these two points.

At which value of $y$ does this line intercept the $y$-axis?

22 December

There are 12 ways of placing 2 tokens on a 2×4 grid so that no two tokens are next to each other horizontally, vertically or diagonally:

O		O

O			O

O
		O

O
			O

	O		O

	O
			O

		O
O

			O
O

			O
	O


O		O


O			O


	O		O

Today's number is the number of ways of placing 2 tokens on a 2×21 grid so that no two tokens are next to each other horizontally, vertically or diagonally.

21 December

Arrange the digits 1–9 (using each digit exactly once) so that the three digit number in:the middle row is a prime number;the bottom row is a square number;the left column is a cube number;the middle column is an odd number;the right column is a multiple of 11.The 3-digit number in the first row is today's number.

			today's number
			prime
			square
cube	odd	multiple of 11

20 December

What is the area of the largest area triangle that has one side of length 32 and one side of length 19?

19 December

The equation $352x^3-528x^2+90=0$ has three distinct real-valued solutions.

Today's number is the number of integers $a$ such that the equation$352x^3-528x^2+a=0$ has three distinct real-valued solutions.

18 December

= 11

= 17

=
11

=
17

17 December

The digital product of a number is computed by multiplying together all of its digits.For example, the digital product of 6273 is 252.

Today's number is the smallest number whose digital product is 252.

16 December

Each clue in this crossnumber is formed of two parts connected by a logical connective:and means that both parts are true;nand means that at most one part is true;or means that at least one part is true;nor means that neither part is true;xor means that exactly one part is true;xnor means that either both parts are false or both parts are true.No number starts with 0.

1	2	3
4
5

1A is a palindrome xnor 1D is a palindrome.1A is greater than 350 nor 1D is less than 150.3D is odd nand 4A and 2D are equal.3D is prime xor 5A is odd.4A is a cube and 2D is a cube.The sum of the digits of 3D is 2 or the sum of the digits of 5A is 5.

Today's number is 1D.

15 December

The odd numbers are written in a pyramid.

(row 1)			1
(row 2)		3		5
(row 3)	7		9		11
	etc.

What is the mean of the numbers in the 19th row?

14 December

You start at the point marked A in the picture below. You want to get to the point marked B. You may travel to the right, upwards, or to the left along the black lines, but you cannot pass along the same line segment more than once.

Today's number is the total number of possible routes to get from A to B.

13 December

The diagram to the left shows three circles and two triangles. The three circles all meet at one point.The vertices of the smaller red triangle are at the centres of the circles. The lines connecting thevertices of the larger blue triangle to the point where all three circles meet are diameters of thethree circles.

The area of the smaller red triangle is 226. What is the area of the larger blue triangle?

12 December

You start at the point marked A in the picture below. You want to get to the point marked B. You may travel to the right or upwards along the black lines.

Today's number is the total number of possible routes to get from A to B.

11 December

The integers are written in a triangle as shown below:

			1
		2	3	4
	5	6	7	8	9
10	11	12	13	14	15	16
etc.

Today's number appears directly above the number 750 in the triangle of integers.

10 December

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct.Today's number is the largest number you can make using the digits in the red boxes.

= 20

= 10

= 15

=
7

=
23

=
15

9 December

Eve writes down a sequence of consecutive positive integers (she writes more than one number). The sum of the numbers Eve has written down is 844.

Today's number is the smallest integer that Eve has written down.

8 December

The sum of three integers is 51. The product of the same three integers is 836. What is the product of largest integer and the second-largest integer?

7 December

The picture below shows eight regular decagons. In each decagon, a red triangle has been drawn with vertices at three of the vertices of the decagon.

The area of each decagon is 240. What is the total area of all the red triangles?

6 December

When 12345 is divided by today's number, the remainder is 205. When 6789 is divided by today's number, the remainder is 112.

5 December

How many different isosceles triangles are there whose perimeter is 50 units, and whose area is an integer number of square-units?

(Two triangles that are rotations, reflections and translations of each other are counted as the same triangle. Triangles with an area of 0 should not be counted.)

4 December

= 5

= 10

=
-6

=
18

=
35

3 December

If you write out the numbers from 1 to 1000 (inclusive), how many times will you write the digit 0?

2 December

The number $7n$ has 37 factors (including 1 and the number itself). How many factors does $8n$ have?

There was a typo in this puzzle. It originally read "38 factors" when it was meant to say "37 factors".

1 December

The geometric mean of a set of $n$ numbers can be computed by multiplying together all the numbers then computing the $n$th root of the result.

The factors of 4 are 1, 2 and 4. The geometric mean of these is 2.

The factors of 6 are 1, 2, 3, and 6. The geometric mean of these is $\sqrt{6}$.

The geometric mean of all the factors of today's number is 22.

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