# Advent Calendar 2016

## Archive

Show me a Random Puzzle**Most Recent Collections**

#### Advent Calendar 2017

#### Sunday Afternoon Maths LXII

What's the Star?#### Sunday Afternoon Maths LXI

XYZ#### Sunday Afternoon Maths LX

Where is Evariste?Bending a Straw

List of All Puzzles

## Tags

time geometry 2d shapes 3d shapes numbers spheres trigonometry complex numbers algebra lines graphs coordinates odd numbers fractions differentiation calculus folding tube maps ellipses triangle numbers money bases triangles squares area square numbers chess probability circles averages speed sport multiples dates factors parabolas functions logic cards games people maths shape prime numbers irreducible numbers probabilty angles proportion dice integration sum to infinity dodecagons hexagons multiplication factorials coins shapes regular shapes colouring grids floors integers rugby crosswords percentages digits sums christmas rectangles clocks menace routes taxicab geometry remainders chalkdust crossnumber palindromes sequences means unit fractions division square roots surds doubling quadratics indices planes volume number partitions ave pascal's triangle mean advent symmetry arrows addition cube numbers star numbers perfect numbers## 1 December

One of the digits of today's number was removed to leave a two digit number.
This two digit number was added to today's number.
The result was 619.

## 2 December

What is the maximum number of lines that can be formed by the intersection
of 30 planes?

## 3 December

What is the volume of the smallest cube inside which a rectangular-based pyramid of volume 266 will fit?

## 4 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 digits in the red boxes.

+ | ÷ | = 2 | |||

+ | ÷ | - | |||

÷ | - | = 5 | |||

÷ | - | × | |||

- | × | = 4 | |||

= 3 | = 5 | = 6 |

## 5 December

Today's number is the number of ways that 35 can be written as the sum of distinct numbers, with none of the numbers in the sum being divisible by 9.

Clarification: By "numbers", I mean (strictly) positive integers. The sum of the same numbers in a different order is counted as the same sum: eg. 1+34 and 34+1 are not different sums.
The trivial sum consisting of just the number 35 counts as a sum.

## 6 December

When you add up the digits of a number, the result is called the digital sum.

How many different digital sums do the numbers from 1 to 10

^{91}have?** There was a mistake in this question (it previously said 10

^{92}). If you answered the typo'd question right, your answer should automatically correct itself to 9 less than it was...## 7 December

Put the digits 1 to 9 (using each digit once) in the boxes so that the three digit numbers formed (reading left to right and top to bottom) have the desired properties written by their rows and columns.

multiple of 25 | |||

today's number | |||

all digits even | |||

multiple of 91 | multiple of 7 | cube number |

## 8 December

Today's number is the second smallest number that can be written as
a×b×c×d×e×f×g×h×i, where
a,b,...,i are all integers greater than 1.

## 9 December

You start at A and are allowed to move either to the right or upwards.

How many different routes are there to get from A to B?

## 10 December

The smallest number that is equal to the sum of its digits
multiplied by ten more than the sum of its digits.

## 11 December

This year, I have spend a lot of time working on the AVE game engine.
Today's answer is the code to the safe in this Christmas themed game.

## 12 December

Here is a list of facts about today's number:

- If a×b is a factor of it, with a and b both positive integers, then either a or b is one.
- The sum of its digits is 14.
- It is odd.
- The product of its digits is 36.
- It is a palindrome when written in base 9.
- It is smaller than yesterday's number.
- It is 4 more than a multiple of 5.
- It is two less than a prime number.
- It is the number of a bus stopping at Richmond station.

## 13 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 smaller number in a red box to the power of the larger number in a red box.

+ | - | = 8 | |||

- | - | - | |||

+ | ÷ | = 3 | |||

+ | ÷ | × | |||

+ | × | = 120 | |||

= 8 | = 1 | = 8 |

## 14 December

In July, I posted the Combining Multiples puzzle.

Today's number is the largest number that cannot be written in the form \(27a+17b\), where \(a\) and \(b\) are positive integers (or 0).

## 15 December

A book has 386 pages. What do the page numbers on the two middle pages add up to?

## 16 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 largest number than can be made from the digits in red boxes.

× | × | = 6 | |||

× | × | × | |||

× | × | = 180 | |||

× | × | × | |||

× | × | = 336 | |||

= 32 | = 70 | = 162 |

## 17 December

The number of degrees in one internal angle of a regular polygon with 360 sides.

## 19 December

The sum of all the numbers in the eighth row of Pascal's triangle.

Clarification: I am starting the counting of rows from 1, not 0. So (1) is the 1st row, (1 1) is the 2nd row, (1 2 1) is the 3rd row, etc.

## 20 December

Earlier this year, I wrote a blog post about different ways to prove Pythagoras' theorem. Today's puzzle uses Pythagoras' theorem.

Start with a line of length 2. Draw a line of length 17 perpendicular to it. Connect the ends to make a right-angled triangle.
The length of the hypotenuse of this triangle will be a non-integer.

Draw a line of length 17 perpendicular to the hypotenuse and make another right-angled triangle. Again the new hypotenuse will have a non-integer length.
Repeat this until you get a hypotenuse of integer length. What is the length of this hypotenuse?

## 21 December

Today's number is a multiple of three. The average (mean) of all the answers that are multiples of three is a multiple of 193.

## 22 December

Today's number is a palindrome. Today's number is also the number of palindromes between 111 and 11111 (including 111 and 11111).

## 23 December

Today's number is the number of three digit numbers that are not three more than a multiple of 7.

## 24 December

Today's number is 191 more than one of the other answers and
100 less than another of the answers.

## 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.