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20210103
It's 2021, and the Advent calendar has disappeared, so it's time to reveal the answers and annouce the winners.
But first, some good news: with your help, Santa received all his letters and Christmas was saved!
Now that the competition is over, the questions and all the answers can be found here.
Before announcing the winners, I'm going to go through some of my favourite puzzles from the calendar, reveal the solution and a couple of other interesting bits and pieces.
Highlights
My first highlight is this puzzle from 5 December, that I think doesn't seem obvious that there's even a unique answer until you spot some nice properties of
dice.
5 December
Carol rolled a large handful of sixsided dice. The total of all the numbers Carol got was 521. After some calculating, Carol worked out that the probability that of her total being 521
was the same as the probability that her total being 200. How many dice did Carol roll?
My next highlight is the puzzle from 15 December. I had a lot of fun trying to write this one. At the end, I tell you that T represents 4: this puzzle still has a unique answer without this,
but it was too hard to get started.
15 December
When talking to someone about this Advent calendar, you told them that the combination of XMAS and MATHS is GREAT.
They were American, so asked you if the combination of XMAS and MATH is great; you said SURE. You asked them their name; they said SAM.
Each of the letters E, X, M, A, T, H, S, R, U, and G stands for a different digit 0 to 9. The following sums are correct:

 
Today's number is SAM. To help you get started, the letter T represents 4.
My next highlight is the puzzle from 16 December. I had a lot of fun writing this one too. At least a few people seem to have enjoyed solving it,
as indicated by this comment.
16 December
Solve the crossnumber to find today's number. No number starts with 0.


My final highlight is the puzzle from 21 December. My method for solving this one was quite complicated. Let me know in the comments or on Twitter if you have a better way of solving it.
21 December
There are 3 ways to order the numbers 1 to 3 so that no number immediately follows the number one less that itself:
 3, 2, 1
 1, 3, 2
 2, 1, 3
Today's number is the number of ways to order the numbers 1 to 6 so that no number immediately follows the number one less that itself.
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 sent
for checking and have looked at these to see which puzzles were the most and least commonly incorrect. The bar chart below shows the total number
of 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 
You can see that the most difficult puzzles were those on
6,
14,
19,
21, and
24 December;
and the easiest puzzles were on
3,
9,
10,
16, and
17 December.
The solutions
The solutions to all the individual puzzles can be found here. If you want to read some alternative solutions to the puzzles, you can find Alira's solutions here, including a very nice explanation of how the final clues fit together to find the directions to Santa's house.
Using the daily clues, it was possible to work out that Santa's house could be found by following the directions 3,7,1,1,4,3,3,4,9,3,3,9 on this map.
Due to the way the Advent compass worked, there are actually a few different ways to get to Santa's house. For example, 7,5,7,1,1,4,3,3,4,9,3,3 also gets you there:
I'll leave you to work out how many different ways there are to get there...
Here's the full collection of tiles that you could find by walking around the map:
The winners
And finally (and maybe most importantly), on to the winners: 223 people found Santa's house entered the competition. This is quite a lot more than in previous years:
2015  2016  2017  2018  2019  2020 
Year 
From the correct answers, the following 10 winners were selected:
1  Mars He 
2  Ben Baker 
3  Amelia Taylor 
4  Alex Ayres 
5  Hannah Charman 
6  Diane Keimel 
7  Alex Burlton 
8  Rob March 
9  Mahmood Hikmet 
10  Guillermo Heras Prieto 
Congratulations! Your prizes will be on their way shortly.
The prizes this year include 2020 Advent calendar Tshirts. If you didn't win one, but would like one of these, I've made them available to buy on Teespring alongside the Tshirts from previous years.
Additionally, well done to
4nder, Ashley Frazer Jarvis, Aaron Stiff, Alan Buck, Alejandro Villarreal, Alex Bolton, Alex Davis, Alex Klapheke, Alexander Rakitin, Anders Kaseorg, Andrew, Andrew P Turner, Andy Ennaco, Annabel, Anthony Della Pella, Artie Smith, Athena, Austin, Austin Shapiro, Beau Ferguson, Becky Russell, Ben Buchwald, Ben Jones, Ben Reiniger, Benjamin Tozer, Bennet, Beth Jensen, Blake, Brennan Dolson, Brian Carnes, Brian Wellington, Burak Kadron, Caleb Likely, Callum Hobbis, Carl Westerlund, Carmen Günther, Cath Simpson, Cathy Hooper, Chris Hellings, Christy Hales, Colin, Colin Brockley, Connie, Corbin Groothuis, Cory Peters, Cosmo Viola, Dan Colestock, Dan DiMillo, Dan McIntyre, Dan Whitman, Dana Sussman, Daniel De la Paz, David, David Ault, David Berardo, David Fox, David Kendel, David Mitchell, David Parmenter, David and Ivy Walbert, Dean Thomas, Deborah Tayler, Don Anderson, Donagh Collins, Elizabeth Blackwell, Emilie Heidenreich, Emily Troyer, Eric Kolbusz, Eric Skoglund, Erik Eklund, Eva, Evan Louis Robinson, Fionn Woodcock, Frances Haigh, Frank Kasell, Franklin Ta, Fred Verheul, Georges, Gabriella Pinter, GertJan de Vries, Graham Greve, Gregory Loges, Gwendal Collet, Harry Allen, Heerpal Sahota, Helen, Helen Matthews, Herschel Pecker, Håkon Balteskard, Ivan Arribas, Jacob Young, James Demers, James Laver, Jan, Jason Demers, Jason Kass, JeanSébastien Turcotte, Jeff Rubin, Jefff Michael, Jen Huang, Jessica Marsh, Jim Rogers, Joe Gage, Johan Asplund, John Alasdair Warwicker, Jon Foster, Jon Lipscombe, Jon Palin, Jonathan Chaffer, Jonathan Forsythe, Jonathan Winfield, Jordan Susskind, Jorge del Castillo, Kai Lam, Karen Climis, Keith Pound, Ken Cheung, Killeen, Kristen Koenigs, Kristin Gramza, Kurt Engleman, Laura Midgley, Lauren Woolsey, Lewis Dyer, Linus BanghartLinn, Liz Madisetti, Louis de Mendonca, Magnus Eklund, Mark Stambaugh, Martijn O., Martin Harris, Martin Holtham, Martine Nome, Matt Askins, Matt Hutton, Matthew Reynier, Matthew Riggle, Matthew Schulz, Matthew Scroggs, Matthias Planitzer, Maxime Rivet, Maximilian Pfister, Michael DeLyser, Michael Prescott, Mihai Zsisku, Mike Hands, Moritz Stocker, Nadine Chaurand, Nathan Chun, Nick C, Nick Keith, Nick Mohr, Norvin Richards, Olov Wilander, Pamela Docherty, Pau Batlle, Paul Livesey, Pranshu Gaba, Ray Arndorfer, Rea, Reuben Cheung, Riccardo Lani, Richard Pemberton, Rileigh Luczak, Road White, Roni, Rosie Paterson, Russ Collins, Ruth Franklin, Ryan Coomer, Ryan Seldon, Ryan Wise, Sam Dreilinger, Sam Hartburn, Sammie Buzzard, Sara Van Hoy, Sarah Brook, Saul Freedman, Scott, Sean, Sean Carmody, Sean Henderson, Sean McDonald, Sebastian, Selina Glauert, Seth Cohen, Shivanshi Adlakha, Simon Schneider, Sophie Maclean, Spencer B., Stephen Cappella, Stephen Dainty, Stephen Jasina, Steve Blay, Steve Paget, Steven Richardson, Susana Early, Tarim, Tim HolmanWilkens, Tim Lewis, Todd Geldon, Tom Fryers, Tony Mann, Tristan Shephard, Valentin Valciu, Vinesh, Yasha, Yuliya Nesterova, Yurie Ito, and Zachary Polansky
who all also found Santa's house but were too unlucky to win prizes this time.
See you all next December, when the Advent calendar will return.
Similar posts
Christmas (2020) is coming!  Christmas (2019) is over  Christmas (2019) is coming!  Christmas (2018) is over 
Comments
Comments in green were written by me. Comments in blue were not written by me.
You can solve the Dec 21 puzzle using the principle of inclusion/exclusion:
There are 6! total ways of arranging 6 numbers.
Now we have to exclude the ones that don't fit. How many ways have 2 following 1? You can think of 12 as a pair, so you're arranging 12/3/4/5/6 in any order, so there are 5! ways to do this. And there are (5 choose 1)=5 total pairs that might exist, so there are 5*5! ways that have either 12, 23, 34, 45, or 56.
Of course, we've double counted some that have more than one pair. (This is where inclusion/exclusion comes in, we have to include them back in). So how many have, say, 12 and 45? Well now we're arranging 12/3/45/6, so there are 4! ways to do so. There are (5 choose 2)=10 different pairs, so the double counting was 10*4!.
We continue this on, and inclusion/exclusion says we keep alternating adding and subtracting as we add more pairs, so the answer is:
6!
 (5 choose 1) * 5!
+ (5 choose 2) * 4!
 (5 choose 3) * 3!
+ (5 choose 4) * 2!
 (5 choose 5) * 1!
= 309
There are 6! total ways of arranging 6 numbers.
Now we have to exclude the ones that don't fit. How many ways have 2 following 1? You can think of 12 as a pair, so you're arranging 12/3/4/5/6 in any order, so there are 5! ways to do this. And there are (5 choose 1)=5 total pairs that might exist, so there are 5*5! ways that have either 12, 23, 34, 45, or 56.
Of course, we've double counted some that have more than one pair. (This is where inclusion/exclusion comes in, we have to include them back in). So how many have, say, 12 and 45? Well now we're arranging 12/3/45/6, so there are 4! ways to do so. There are (5 choose 2)=10 different pairs, so the double counting was 10*4!.
We continue this on, and inclusion/exclusion says we keep alternating adding and subtracting as we add more pairs, so the answer is:
6!
 (5 choose 1) * 5!
+ (5 choose 2) * 4!
 (5 choose 3) * 3!
+ (5 choose 4) * 2!
 (5 choose 5) * 1!
= 309
Todd
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20201203
In November, I spent some time designing this year's Chalkdust puzzle Christmas card.
The card looks boring at first glance, but contains 9 puzzles. By splitting the answers into two digit numbers, then colouring the regions labelled with each number (eg if an answer to a question in the red section is 201304, colour the regions labelled 20, 13 and 4 red), you will reveal a Christmas themed picture.
If you want to try the card yourself, you can download this pdf. Alternatively, you can find the puzzles below and type the answers in the boxes. The answers will be automatically be split into two digit numbers, and the regions will be coloured...
Grey/black  
1.  How many odd numbers can you make (by writing digits next to each other, so 13, 1253, and 457 all count) using the digits 1, 2, 3, 4, 5, and 7 each at most once (and no other digits)?  Answer 
2.  Carol made a book by stacking 40300 pieces of paper, folding the stack in half, then writing the numbers 1 to 161200 on the pages. She then pulled out one piece of paper and added up the four numbers written on it. What is the largest number she could have reached?  Answer 
3.  What is the sum of all the odd numbers between 0 and 130376?  Answer 
White/yellow  
4.  There are three cards with integers written on them. The pairs of cards add to 31, 35 and 36. What is the sum of all three cards?  Answer 
5.  What is the volume of the smallest cuboid that a squarebased pyramid with volume 1337 can fit inside??  Answer 
6.  What is the lowest common multiple of 305 and 671?  Answer 
Red  
7.  Holly rolled a huge pile of dice and added up all the top faces to get 6136. She realised that the probability of getting 6136 was the same as getting 9999. How many dice did she roll?  Answer 
8.  How many squares (of any size) are there in a 14×16 grid of squares?  Answer 
9.  Ivy picked a number, removed a digit, then added her two numbers to get 155667. What was her original number?  Answer 
Similar posts
Christmas card 2019  Christmas card 2018  Christmas card 2017  Christmas card 2016 
Comments
Comments in green were written by me. Comments in blue were not written by me.
@JDev: lots of the card will still be brown once you're done, but you should see a nice picture. Perhaps one of your answers is wrong, making a mess of the picture?
Matthew
I finished all of the puzzles but the picture is far from colored in. Am I missing something?
These puzzles have been a blast!
These puzzles have been a blast!
JDev
@Tara: I initially made the same mistake. Maybe you didn't take into account that 6 is not one of the available digits in question 1?
Sean
@Tara: Yes, looks like you may have got an incorrect answer for one of the black puzzles
Matthew
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20201122
This year, the front page of mscroggs.co.uk will once again feature an Advent calendar, just like
in the five previous Decembers.
Behind each door, there will be a puzzle with a three digit solution. The solution to each day's puzzle forms part of a logic puzzle:
It's nearly Christmas and something terrible has happened: you've just landed in a town in the Arctic circle with a massive bag of letters for Santa, but you've lost to instructions for how to get to Santa's house near the north pole. You need to work out where he lives and deliver the letters to him before Christmas is ruined for everyone.
Due to magnetic compasses being hard to use near the north pole, you brought with you a special Advent compass. This compass has nine numbered directions. Santa has given the residents of the town clues about a sequence of directions that will lead to his house; but in order to keep his location secret from present thieves, he gave each resident two clues: one clue is true, and one clue is false.
The residents' clues will reveal to you a seqeunce of compass directions to follow. You can try out your sequences on this map.
Behind each day (except Christmas Day), there is a puzzle with a threedigit answer. Each of these answers forms part of a resident's clue. You must use these clues to work out how to find Santa's house.
Ten randomly selected people who solve all the puzzles, find Santa's house, and fill in the entry form behind the door on the 25th will win prizes!
The winners will be randomly chosen from all those who submit the entry form before the end of 2020. Each day's puzzle (and the entry form on Christmas Day) will be available from 5:00am GMT. But as the winners will be selected randomly,
there's no need to get up at 5am on Christmas Day to enter!
As you solve the puzzles, your answers will be stored. To share your stored answers between multiple devices, enter your email address below the calendar and you will be emailed a magic link to visit on your other devices.
To win a prize, you must submit your entry before the end of 2020. Only one entry will be accepted per person. If you have any questions, ask them in the comments below or on Twitter.
So once December is here, get solving! Good luck and have a very merry Christmas!
Similar posts
Christmas (2020) is over  Christmas (2019) is over  Christmas (2019) is coming!  Christmas (2018) is over 
Comments
Comments in green were written by me. Comments in blue were not written by me.
Nice one today (16 December) :)
Did he?... did he really?... starts to look like it... yes he did! :D
Did he?... did he really?... starts to look like it... yes he did! :D
GertJan
@Dean: you can also go through each answer one at a time and change a digit; if the number of wrong answers goes up then your answer for that question was correct
A
@Dean: Yes, I'm planning to change how that bit works. Check back tomorrow or the next day for a more fun finish!
Matthew
A bit harsh that we can’t tell exactly which answers are wrong! I won’t have time to revisit every puzzle  and its kind of less fun redoing something that is already correct... :(
Dean
@Marty: Yes, I'm in the middle of correcting the clues page to add these details back
Matthew
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20200102
It's 2020, and the Advent calendar has disappeared, so it's time to reveal the answers and annouce the winners.
But first, some good news: with your help, Santa and his two reindeer were found and Christmas was saved!
Now that the competition is over, the questions and all the answers can be found here.
Before announcing the winners, I'm going to go through some of my favourite puzzles from the calendar, reveal the solution and a couple of notes and Easter eggs.
Highlights
My first highlight is this puzzle from 10 December that seems difficult to get started on, but plotting the two quadratics reveals a very useful piece of information that can be used.
10 December
For all values of \(x\), the function \(f(x)=ax+b\) satisfies
$$8x8x^2\leqslant f(x)\leqslant x^2.$$
What is \(f(65)\)?
Edit: The lefthand quadratic originally said \(88xx^2\). This was a typo and has now been corrected.
My next highlight is the puzzle from 12 December, which was election day in the UK. Although the puzzle isn't that difficult or interesting to calculate, the answer is surprising enough
to make this one of my favourites.
12 December
For a general election, the Advent isles are split into 650 constituencies. In each constituency, exactly 99 people vote: everyone votes for one of the two main parties: the Rum party or the
Land party. The party that receives the most votes in each constituency gets an MAP (Member of Advent Parliament) elected to parliament to represent that constituency.
In this year's election, exactly half of the 64350 total voters voted for the Rum party. What is the largest number of MAPs that the Rum party could have?
My next highlight is the puzzle from 13 December. If you enjoyed this one, then you can find a puzzle based on a similar idea on the puzzles
pages of issue 10 of Chalkdust.
13 December
Each clue in this crossnumber (except 5A) gives a property of that answer that is true of no other answer. For example: 7A is a multiple of 13; but 1A, 3A, 5A, 1D, 2D, 4D, and 6D are all not multiples of 13.
No number starts with 0.


My final highlight is the puzzle from 16 December. I always include a few of these, as they can be designed to give any answer so are useful for making the final logic puzzle work.
But I was particularly happy with this one.
16 December
Arrange the digits 19 in a 3×3 square so that:
the median number in the first row is 6;
the median number in the second row is 3;
the mean of the numbers in the third row is 4;
the mean of the numbers in the second column is 7;
the range of the numbers in the third column is 2,
The 3digit number in the first column is today's number.
median 6  
median 3  
mean 4  
today's number  mean 7  range 2 
Notes and Easter eggs
A few of you may have noticed the relevance of the colours assigned to each island:
Rum (Red Rum),
Land (Greenland),
Moon (Blue Moon), and
County (Orange County).
Once you've entered 24 answers, the calendar checks these and tells you how many are correct. I logged the answers that were sent
for checking and have looked at these to see which puzzles were the most and least commonly incorrect. The bar chart below shows the total number
of 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 
You can see that the most difficult puzzles were those on
4 and
15 December;
and the easiest puzzles were on
3,
11,
17, and
24 December.
This year, the final logic puzzle revealed the positions of Santa, Rudolph and Blitzen, then you had to find them on this map.
The map has 6 levels, with 81 positions on each level, so the total size of the map is \(81^6=282\,429\,536\,481\) squares.
This is a lot; one Advent solver even wondered how large a cross stitch of the whole thing would be.
I obviously didn't draw 282 billion tiles: the whole map was generated using the following tiles, which were reused across the map.
I also snuck a small Easter egg into the map. Below the Advent calendar, this example was shown:
If you actually visited this position on the map, you found Wally.
At least one Advent solver appears to have found Wally, as they left this cryptic comment under the name Dr Matrix
(an excellent Martin Gardner reference).
The solution
The solutions to all the individual puzzles can be found here.
Using the daily clues, it was possible to work out that
Santa was at 36.11.19.79.79.43, Blitzen was at 23.12.23.11.23.11, and Rudolph was at 16.64.16.16.16.64.
And finally (and maybe most importantly), on to the winners: 126 people found Santa and both reindeer and entered the competition. Their (very) approximate locations are shown on this map:
From the correct answers, the following 10 winners were selected:
1  Valentin Valciu 
2  Tom Anderson 
3  Alex Bolton 
4  Kevin Fray 
5  Jack Kiuttu 
6  Ben Baker 
7  Joe Gage 
8  Michael Thomas 
9  Martin Holtham 
10  Beth Jensen 
Congratulations! Your prizes will be on their way shortly.
The prizes this year include 2019 Advent calendar Tshirts. If you didn't win one, but would like one of these, you can buy one at mscroggs.co.uk/tshirt until 7 January
(when I will be sending them for printing).
Additionally, well done to
Adam Abrams, Adam NH, Adam Vellender, Alan Buck, Alex Ayres, Alex Burlton, Alexander Ignatenkov, Andrew Ennaco, Andrew Tindall, Artie Smith, Ashley Jarvis, Austin Antoniou, Becky Russell,
Ben Jones, Ben Reiniger, Brennan Dolson, Brian Carnes, Carl Westerlund, Carmen Guenther, Clare Wallace, Colin Beveridge, Connie, Corbin Groothuis, Cory Peters, Dan Colestock, Dan DiMillo,
Dan Whitman, David, David Ault, David Fox, Diane Keimel, Duncan Ramage, Emilie Heidenreich, Emily Troyer, Eric, Eric Kolbusz, Erik Eklund, Evan Louis Robinson, Frances Haigh, Franklin Ta,
Fred Verheul, Félix Breton, Gabriella Pinter, Gautam Webb, GertJan de Vries, Hart Andrin, Heerpal Sahota, Herschel Pecker, Jacob Juillerat, Jan, JeanNoël Monette, Jen Shackley,
Jeremiah Southwick, Jessica Marsh, Johan Asplund, John Warwicker, Jon Foster, Jon Palin, Jonathan Chaffer, Jonathan Winfield, Jose, Kai, Karen Climis, Karen Kendel, Katja Labermeyer, Laura,
Lewis Dyer, Louis de Mendonca, M Oostrom, Magnus Eklund, Mahmood Hikmet, Marian Clegg, Mark Stambaugh, Martin Harris, Martine Vijn Nome, Matt Hutton, Matthew Askins, Matthew Schulz,
Maximilian Pfister, Melissa Lucas, Mels, Michael DeLyser, Michael Gustin, Michael Horst, Michael Prescott, Mihai Zsisku, Mike, Mikko, Moreno Gennari, Nadine Chaurand, Naomi Bowler, Nathan C,
Pat Ashforth, Paul Livesey, Pranshu Gaba, Raymond Arndorfer, Riccardo Lani, Rosie Paterson, Rupinder Matharu, Russ Collins, S A Paget, SShaikh, Sam Butler, Sam Hartburn, Scott, Seth Cohen,
Shivanshi Adlakha, Simon Schneider, Stephen Cappella, Stephen Dainty, Steve Blay, Thomas Tu, Tom Anderson, Tony Mann, Yasha, and Yuliya Nesterova,
who all also submitted the correct answer but were too unlucky to win prizes this time.
See you all next December, when the Advent calendar will return.
Similar posts
Christmas (2020) is over  Christmas (2020) is coming!  Christmas (2019) is coming!  Christmas (2018) is over 
Comments
Comments in green were written by me. Comments in blue were not written by me.
It's interesting that the three puzzles with the most incorrect attempts can all be looked up on OEIS.
Day 15  https://oeis.org/A001055  "number of ways of factoring n with all factors greater than 1"
Day 4  https://oeis.org/A001045  "number of ways to tile a 2 × (n1) rectangle with 1 × 2 dominoes and 2 × 2 squares"
Day 2  https://oeis.org/A002623  "number of nondegenerate triangles that can be made from rods of length 1,2,3,4,...,n"
Day 15  https://oeis.org/A001055  "number of ways of factoring n with all factors greater than 1"
Day 4  https://oeis.org/A001045  "number of ways to tile a 2 × (n1) rectangle with 1 × 2 dominoes and 2 × 2 squares"
Day 2  https://oeis.org/A002623  "number of nondegenerate triangles that can be made from rods of length 1,2,3,4,...,n"
Adam Abrams
Never mind, I found them, they were your example ASC. Very clever!
Michael T
Where did the coordinates for Wally come from? Are they meaningful, or are they just some random coordinates thrown together?
Michael T
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20191208
Just like last year, the year before and the year before, TD and I spent some time in November this year designing a Chalkdust puzzle Christmas card.
The card looks boring at first glance, but contains 9 puzzles. By splitting the answers into two digit numbers, then drawing lines labelled with each number (eg if an answer is 201304, draw the lines labelled 20, 13 and 4), you will reveal a Christmas themed picture. Colouring the regions of the card containing circles red, the regions containing squares green, and the regions containing stars white or yellow will make this picture even nicer.
If you want to try the card yourself, you can download this pdf. Alternatively, you can find the puzzles below and type the answers in the boxes. The answers will be automatically be split into two digit numbers, the lines will be drawn, and the regions will be coloured...
1.  If you write out the numbers from 1 to 10,000 (inclusive), how many times will you write the digit 1?  Answer 
2.  What is the sum of all the odd numbers between 0 and 86?  Answer 
3.  How many numbers between 1 and 4,008,004 (inclusive) have an odd number of factors (including 1 and the number itself)?  Answer 
4.  In a book with pages numbered from 1 to 130,404, what do the two page numbers on the centre spread add up to?  Answer 
5.  What is the area of the largest area quadrilateral that will fit inside a circle with area 60,153π?  Answer 
6.  There are 5 ways to write 4 as the sum of ones and twos (1+1+1+1, 1+1+2, 1+2+1, 2+1+1, and 2+2). How many ways can you write 28 as the sum of ones and twos?  Answer 
7.  What is the lowest common multiple of 1025 and 835?  Answer 
8.  How many zeros does 245! (245!=245×244×243×...×1) end in?  Answer 
9.  Carol picked a 6digit number then removed one of its digits to make a 5digit number. The sum of her 6digit and 5digit numbers is 334877. Which 6digit number did she pick?  Answer 
Similar posts
Christmas card 2020  Christmas card 2018  Christmas card 2017  Christmas card 2016 
Comments
Comments in green were written by me. Comments in blue were not written by me.
Thanks for the feedback. (I now understand the need for redaction). My son sent me your link as a Xmas present. I must think of an appropriate retaliation. (What is a PDF?)Think I've fixed 1,6 and 9....8 eludes me, for the moment.
Rob
@Rob: It looks to me like you've made mistakes in questions 1, 6, 8, and 9. The hints from the back of the pdf might help:
1. How many numbers between 1 and 10,000 have 1 as their final digit? How many have 1 as their penultimate digit?
6. How many ways can you write 1? 2? 3? 4? 5? What's the pattern?
8. How many zeros does 10! end in? How many zeros does 20! end in? How many zeros does 30! end in?
9. Carol’s sum is odd. What does this tell you about the 5 and 6digit numbers?
1. How many numbers between 1 and 10,000 have 1 as their final digit? How many have 1 as their penultimate digit?
6. How many ways can you write 1? 2? 3? 4? 5? What's the pattern?
8. How many zeros does 10! end in? How many zeros does 20! end in? How many zeros does 30! end in?
9. Carol’s sum is odd. What does this tell you about the 5 and 6digit numbers?
Matthew
I'm 71, with one good eye left. What am I missing?
1. 400001
2. 1849
3. 2002
4. 130405
5. 120306
6. 53?
7. 171175
8. 59?
9. 313525
1. 400001
2. 1849
3. 2002
4. 130405
5. 120306
6. 53?
7. 171175
8. 59?
9. 313525
Rob
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The 16th was just cheeky, after I spent way to much time on the 15th it was nice to have something like that!
It's been 16+ years since I did any probability or combinations and permutations, so it was nice to brush off that part of my brain, not that I did any of them well, but it should serve me well when my kids start doing them and ask me for help.