# Puzzles

## Archive

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#### Sunday Afternoon Maths LXVII

Coloured weightsNot Roman numerals

#### Advent calendar 2018

#### Sunday Afternoon Maths LXVI

Cryptic crossnumber #2#### Sunday Afternoon Maths LXV

Cryptic crossnumber #1Breaking Chocolate

Square and cube endings

List of all puzzles

## Tags

quadratics hexagons coins addition unit fractions books dates people maths area proportion integers graphs parabolas folding tube maps logic 3d shapes number dodecagons partitions spheres angles remainders bases factors advent cryptic clues factorials averages dice taxicab geometry pascal's triangle planes numbers calculus algebra probabilty floors means regular shapes cryptic crossnumbers sum to infinity routes geometry clocks menace circles colouring symmetry chalkdust crossnumber functions multiplication ellipses percentages doubling grids star numbers lines balancing perimeter speed complex numbers squares digits christmas scales arrows differentiation rectangles triangles shapes sport sums games multiples triangle numbers odd numbers square numbers division shape cube numbers palindromes chocolate polygons ave money mean crosswords rugby trigonometry fractions crossnumbers integration surds 2d shapes perfect numbers volume probability irreducible numbers sequences prime numbers chess time square roots cards coordinates wordplay indices## Advent 2018 Logic Puzzle

2018's Advent calendar ended with a logic puzzle: It's nearly Christmas and something terrible has happened: one of Santa's five helpers—Kip Urples, Bob Luey, Jo Ranger, Fred Metcalfe, and Meg Reeny—has stolen all the presents during the North Pole's annual Sevenstival. You need to find the culprit before Christmas is ruined for everyone.

Every year in late November, Santa is called away from the North Pole for a ten hour meeting in which a judgemental group of elders decide who has been good and who has been naughty. While Santa is away, it is traditional for his helpers celebrate Sevenstival.
Sevenstival gets in name from the requirement that every helper must take part in exactly seven activities during the celebration; this year's
available activities were billiards, curling, having lunch, solving maths puzzles, table tennis, skiing, chess, climbing and ice skating.

Each activity must be completed in one solid block: it is forbidden to spend some time doing an activity, take a break to do something else then return to the first activity.
This year's Sevenstival took place between 0:00 and 10:00 (North Pole standard time).

During this year's Sevenstival, one of Santa's helpers seven activities included stealing all the presents from Santa's workshop.
Santa's helpers have 24 pieces of information to give to you, but the culprit is going to lie about everything in an attempt to confuse you, so be careful who you trust.

Here are the clues:

1

Meg says: "Between

Meg says: "Between

**2:33**and curling, I played billiards with Jo."15

Kip says: "The curling match lasted

Kip says: "The curling match lasted

**323**mins."24

Fred says: "In total, Jo and Meg spent

Fred says: "In total, Jo and Meg spent

**1**hour and**57**mins having lunch."8

Meg says: "A total of

Meg says: "A total of

**691**mins were spent solving maths puzzles."17

Jo says: "I played table tennis with Fred and Meg for

Jo says: "I played table tennis with Fred and Meg for

**2**+**8**+**5**mins."23

Meg says: "

Meg says: "

**1:32**was during my 83 min ski"7

Meg says: "The number of mins the curling game lasted is a factor of

Meg says: "The number of mins the curling game lasted is a factor of

**969**."16

Jo says: "I started skiing with Bob, and finished before Bob at

Jo says: "I started skiing with Bob, and finished before Bob at

**8:45**."5

Jo says: "At

Jo says: "At

**4:45**, Fred, Bob, Kip and I started a curling match."14

Fred says: "I spent

Fred says: "I spent

**135**mins playing chess with Meg."20

Meg says: "Jo started skiing at

Meg says: "Jo started skiing at

**7:30**."4

Bob says: "I went for a

Bob says: "I went for a

**150**min ski."13

Kip says: "Jo started skiing at

Kip says: "Jo started skiing at

**6:08**."22

Fred says: "Bob, Kip and I finished lunch at

Fred says: "Bob, Kip and I finished lunch at

**3:30**."6

Bob says: "I played billiards with Kip from 0:00 until

Bob says: "I played billiards with Kip from 0:00 until

**1:21**."12

Fred says: "Between 3:30 and 4:45, there were

Fred says: "Between 3:30 and 4:45, there were

**3**people climbing."21

Fred says: "In total, Bob, Meg and I spent

Fred says: "In total, Bob, Meg and I spent

**269**mins ice skating."10

Meg says: "Between 0:00 and

Meg says: "Between 0:00 and

**1:10**, I was ice skating."19

Jo says: "At

Jo says: "At

**1:12**, Fred and I were both in the middle of maths puzzles."3

Jo says: "Straight after curling, I had a

Jo says: "Straight after curling, I had a

**108**min game of chess with Kip."9

Fred says: "At

Fred says: "At

**2:52**, I started having lunch with Bob and Kip."18

Jo says: "I spent

Jo says: "I spent

**153**mins solving maths puzzles."2

Fred says: "I was solving maths puzzles for

Fred says: "I was solving maths puzzles for

**172**mins."11

Meg says: "I spent

Meg says: "I spent

**108**mins solving maths puzzles with Bob."## Advent 2017 logic puzzle

2017's Advent calendar ended with a logic puzzle: It's nearly Christmas and something terrible has happened: Santa and his two elves have been cursed! The curse has led Santa to forget which present three childrenâ€”Alex, Ben and Carolâ€”want and where they live.

The elves can still remember everything about Alex, Ben and Carol, but the curse is causing them to lie. One of the elves will lie on even numbered days and tell the truth on odd numbered days; the other elf will lie on odd numbered days and tell the truth on even numbered days. As is common in elf culture, each elf wears the same coloured clothes every day.

Each child lives in a different place and wants a different present. (But a present may be equal to a home.) The homes and presents are each represented by a number from 1 to 9.

Here are the clues:

21

White shirt says: "Yesterday's elf lied: Carol wants

White shirt says: "Yesterday's elf lied: Carol wants

**4**,**9**or**6**."10

Orange hat says: "

Orange hat says: "

**249**is my favourite number."5

Red shoes says: "Alex lives at

Red shoes says: "Alex lives at

**1**,**9**or**6**."16

Blue shoes says: "I'm the same elf as yesterday. Ben wants

Blue shoes says: "I'm the same elf as yesterday. Ben wants

**5**,**7**or**0**."23

Red shoes says: "Carol wants a factor of

Red shoes says: "Carol wants a factor of

**120**. I am yesterday's elf."4

Blue shoes says: "

Blue shoes says: "

**495**is my favourite number."15

Blue shoes says: "Carol lives at

Blue shoes says: "Carol lives at

**9**,**6**or**8**."22

Purple trousers says: "Carol wants a factor of

Purple trousers says: "Carol wants a factor of

**294**."11

White shirt says: "

White shirt says: "

**497**is my favourite number."6

Pink shirt says: "Ben does not live at the last digit of

Pink shirt says: "Ben does not live at the last digit of

**106**."9

Blue shoes says: "Ben lives at

Blue shoes says: "Ben lives at

**5**,**1**or**2**."20

Orange hat says: "Carol wants the first digit of

Orange hat says: "Carol wants the first digit of

**233**."1

Red shoes says: "Alex wants

Red shoes says: "Alex wants

**1**,**2**or**3**."24

Green hat says: "The product of the six final presents and homes is

Green hat says: "The product of the six final presents and homes is

**960**."17

Grey trousers says: "Alex wants the first digit of

Grey trousers says: "Alex wants the first digit of

**194**."14

Pink shirt says: "One child lives at the first digit of

Pink shirt says: "One child lives at the first digit of

**819**."3

White shirt says: "Alex lives at

White shirt says: "Alex lives at

**2**,**1**or**6**."18

Green hat says: "Ben wants

Green hat says: "Ben wants

**1**,**5**or**4**."7

Green hat says: "Ben lives at

Green hat says: "Ben lives at

**3**,**4**or**3**."12

Grey trousers says: "Alex lives at

Grey trousers says: "Alex lives at

**3**,**1**or**5**."19

Purple trousers says: "Carol lives at

Purple trousers says: "Carol lives at

**2**,**6**or**8**."8

Red shoes says: "The digits of

Red shoes says: "The digits of

**529**are the toys the children want."13

Green hat says: "One child lives at the first digit of

Green hat says: "One child lives at the first digit of

**755**."2

Red shoes says: "Alex wants

Red shoes says: "Alex wants

**1**,**4**or**2**."## 23 December

In the song

*The Twelve Days of Christmas*, how many presents have been given after 8 days?## What's the star?

In the Christmas tree below, the rectangle, baubles, and the star at the top each contain a number. The square baubles contain square numbers; the triangle baubles contain triangle numbers; and the cube bauble contains a cube number.

The numbers in the rectangles (and the star) are equal to the sum of the numbers below them. For example, if the following numbers are filled in:

then you can deduce the following:

What is the number in the star at the top of this tree?

*You can download a printable pdf of this puzzle here.*

## Advent 2016 murder mystery

2016's Advent calendar ended with a murder mystery, with each of
the murderer, motive, weapon and location being a digit from 1 to 9.
Here are the clues:

10

None of the digits of

None of the digits of

**171**is the location.3

None of the digits of

None of the digits of

**798**is the motive.7

One of the digits of

One of the digits of

**691**is the location.16

None of the digits of

None of the digits of

**543**is the location.5

One of the digits of

One of the digits of

**414**is the murderer.20

The first digit of

The first digit of

**287**is the number of false red clues.8

Clues on days that are factors of

Clues on days that are factors of

**768**are all true.22

The murderer is the square root of one of the digits of

The murderer is the square root of one of the digits of

**191**.11

One of the digits of

One of the digits of

**811**is the weapon.19

The highest common factor of the weapon and

The highest common factor of the weapon and

**128**is 1.13

None of the digits of

None of the digits of

**512**is the murderer.18

One of the digits of

One of the digits of

**799**is the motive.17

None of the digits of

None of the digits of

**179**is the motive.6

None of the digits of

None of the digits of

**819**is the location.24

One of the digits of

One of the digits of

**319**is total number of false clues.23

One of the digits of

One of the digits of

**771**is the murderer.2

The weapon is not one of the digits of

The weapon is not one of the digits of

**435**.14

The final digit of

The final digit of

**415**is the number of true blue clues.4

The weapon is a factor of

The weapon is a factor of

**140**.12

The number of false clues before today is the first digit of

The number of false clues before today is the first digit of

**419**.9

One of the digits of

One of the digits of

**447**is the motive.1

None of the digits of

None of the digits of

**563**is the motive.21

One of the digits of

One of the digits of

**816**is the murderer.15

One of the digits of

One of the digits of

**387**is the motive.## Santa

Each of the letters D, A, Y, S, N, T, B, R and E represents a different non-zero digit. The following sum is true:

$$
\begin{array}{cccccc}
D&A&D&D&Y\\
B&E&A&R&D&+\\
\hline
S&A&N&T&A
\end{array}
$$
This has a unique solution, but I haven't found a way to find the solution without brute force. This less insightful sum is also true with the same values of the letters (and should allow you to find the values of the letters using logic alone):

$$
\begin{array}{ccccc}
R&A&T&S\\
N&E&R&D&+\\
\hline
S&A&N&E
\end{array}
$$