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mscroggs.co.uk subscribe 2018's Advent calendar ended with a logic puzzle: It's nearly Christmas and something terrible has happened: one of Santa's five helpers—Bob Luey, Fred Metcalfe, Jo Ranger, Kip Urples, 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 2:33 and curling, I played billiards with Jo."
15
Kip says: "The curling match lasted 323 mins."
24
Fred says: "In total, Jo and Meg spent 1 hour and 57 mins having lunch."
8
Meg says: "A total of 691 mins were spent solving maths puzzles."
17
Jo says: "I played table tennis with Fred and Meg for 2+8+5 mins."
23
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 969."
16
Jo says: "I started skiing with Bob, and finished before Bob at 8:45."
5
Jo says: "At 4:45, Fred, Bob, Kip and I started a curling match."
14
Fred says: "I spent 135 mins playing chess with Meg."
20
Meg says: "Jo started skiing at 7:30."
4
Bob says: "I went for a 150 min ski."
13
Kip says: "Jo started skiing at 6:08."
22
Fred says: "Bob, Kip and I finished lunch at 3:30."
6
Bob says: "I played billiards with Kip from 0:00 until 1:21."
12
Fred says: "Between 3:30 and 4:45, there were 3 people climbing."
21
Fred says: "In total, Bob, Meg and I spent 269 mins ice skating."
10
Meg says: "Between 0:00 and 1:10, I was ice skating."
19
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 108 min game of chess with Kip."
9
Fred says: "At 2:52, I started having lunch with Bob and Kip."
18
Jo says: "I spent 153 mins solving maths puzzles."
2
Fred says: "I was solving maths puzzles for 172 mins."
11
Meg says: "I spent 108 mins solving maths puzzles with Bob."

## 24 December 1,0,2,0,1,1
The sequence of six numbers above has two properties:
1. Each number is either 0, 1 or 2.
2. Each pair of consecutive numbers adds to (strictly) less than 3.
Today's number is the number of sequences of six numbers with these two properties
Tags: numbers

## 23 December Today's number is the area of the largest area rectangle with perimeter 46 and whose sides are all integer length.

## 22 December In base 2, 1/24 is 0.0000101010101010101010101010...
In base 3, 1/24 is 0.0010101010101010101010101010...
In base 4, 1/24 is 0.0022222222222222222222222222...
In base 5, 1/24 is 0.0101010101010101010101010101...
In base 6, 1/24 is 0.013.
Therefore base 6 is the lowest base in which 1/24 has a finite number of digits.
Today's number is the smallest base in which 1/10890 has a finite number of digits.
Note: 1/24 always represents 1 divided by twenty-four (ie the 24 is written in decimal).

## 21 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 smallest number you can make using the digits in the red boxes.
 + ÷ = 2 × + - × - = 31 + + - - × = 42 =37 =13 =-2
Tags: numbers, grids

## 20 December Today's number is the sum of all the numbers less than 40 that are not factors of 40.

## 19 December Today's number is the number of 6-dimensional sides on a 8-dimensional hypercube.
Tags: 3d shapes

## 18 December There are 6 terms in the expansion of $$(x+y+z)^2$$:
$$(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2xz$$
Today's number is number of terms in the expansion of $$(x+y+z)^{16}$$.
Tags: algebra

## 17 December For $$x$$ and $$y$$ between 1 and 9 (including 1 and 9), I write a number at the co-ordinate $$(x,y)$$: if $$x\lt y$$, I write $$x$$; if not, I write $$y$$.
Today's number is the sum of the 81 numbers that I have written.
Tags: numbers

## 16 December Arrange the digits 1-9 in a 3×3 square so that the first row makes a triangle number, the second row's digits are all even, the third row's digits are all odd; the first column makes a square number, and the second column makes a cube number. The number in the third column is today's number.
 triangle all digits even all digits odd square cube today's number
Tags: numbers, grids

## 15 December Today's number is smallest three digit palindrome whose digits are all non-zero, and that is not divisible by any of its digits.

## 14 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.
 - + = 10 ÷ + ÷ ÷ + = 3 + - ÷ + × = 33 =7 =3 =3
Tags: numbers, grids

## 13 December There is a row of 1000 lockers numbered from 1 to 1000. Locker 1 is closed and locked and the rest are open.
A queue of people each do the following (until all the lockers are closed):
• Close and lock the lowest numbered locker with an open door.
• Walk along the rest of the queue of lockers and change the state (open them if they're closed and close them if they're open) of all the lockers that are multiples of the locker they locked.
Today's number is the number of lockers that are locked at the end of the process.
Note: closed and locked are different states.

## 12 December There are 2600 different ways to pick three vertices of a regular 26-sided shape. Sometime the three vertices you pick form a right angled triangle. These three vertices form a right angled triangle.
Today's number is the number of different ways to pick three vertices of a regular 26-sided shape so that the three vertices make a right angled triangle.

## 11 December This puzzle is inspired by a puzzle Woody showed me at MathsJam.
Today's number is the number $$n$$ such that $$\frac{216!\times215!\times214!\times...\times1!}{n!}$$ is a square number.

## 10 December The equation $$x^2+1512x+414720=0$$ has two integer solutions.
Today's number is the number of (positive or negative) integers $$b$$ such that $$x^2+bx+414720=0$$ has two integer solutions.

## 9 December Today's number is the number of numbers between 10 and 1,000 that contain no 0, 1, 2 or 3.

## 8 December Arrange the digits 1-9 in a 3×3 square so: each digit the first row is the number of letters in the (English) name of the previous digit, each digit in the second row is one less than the previous digit, each digit in the third row is a multiple of the previous digit; the second column is an 3-digit even number, and the third column contains one even digit. The number in the first column is today's number.
 each digit is the number of letters in the previous digit each digit is one less than previous each digit is multiple of previous today's number even 1 even digit
Edit: There was a mistake in this puzzle: the original had two solutions. If you entered the wrong solution, it will automatically change to the correct one.
Tags: numbers, grids

## 7 December There is a row of 1000 closed lockers numbered from 1 to 1000 (inclusive). Near the lockers, there is a bucket containing the numbers 1 to 1000 (inclusive) written on scraps of paper.
1000 people then each do the following:
• Pick a number from the bucket (and don't put it back).
• Walk along the row of lockers and change the state (open them if they're closed and close them if they're open) of all the lockers that are multiples of the number they picked (including the number they picked).
Today's number is the number of lockers that will be closed at the end of this process.

## 6 December This puzzle is inspired by a puzzle that Daniel Griller showed me.
Write down the numbers from 12 to 22 (including 12 and 22). Under each number, write down its largest odd factor*.
Today's number is the sum of all these odd factors.
* If a number is odd, then its largest odd factor is the number itself.

## 5 December I make a book by taking 111 sheets of paper, folding them all in half, then stapling them all together through the fold. I then number the pages from 1 to 444.
Today's number is the sum of the two page numbers on the centre spread of my book.
Tags: numbers, books

## 4 December Today's number is the number of 0s that 611! (611×610×...×2×1) ends in.

## 3 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 - + × + - = 11 - - - + + = 11 =-11 =11 =11
Tags: numbers, grids

## 2 December Today's number is the area of the largest dodecagon that it's possible to fit inside a circle with area $$\displaystyle\frac{172\pi}3$$.

## 1 December There are 5 ways to write 4 as the sum of 1s and 2s:
• 1+1+1+1
• 2+1+1
• 1+2+1
• 1+1+2
• 2+2
Today's number is the number of ways you can write 12 as the sum of 1s and 2s.
Tags: numbers

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