Advent calendar 2025
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 all
the 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.
10
The chassis should not be ordered from 918
The chassis should not be ordered from 712
One of the parts should be ordered from 28
The chassis should not be ordered from 49
The front should not be ordered from 51
The chassis should not be ordered from 67
Santa's seat should be ordered from 3, 5, or 46
The front should not be ordered from 814
One of the parts should be ordered from 413
The runners should not be ordered from a factor of 17520
Today's clue and the clue on 19 December are false4
The chassis should not be ordered from 52
The chassis should not be ordered from 15
The chassis should not be ordered from 324
Santa's seat should be ordered from 6, 7, or 517
The runners should be ordered from 511
The chassis should not be ordered from 421
One of the parts should be ordered from 83
The front should not be ordered from 119
Exactly 4 of the clues are false15
The chassis should not be ordered from 1, 2, or 823
The chassis should be ordered from 422
The chassis should be ordered from 416
One of the parts should be ordered from 110
The chassis should not be ordered from 918
The chassis should not be ordered from 712
One of the parts should be ordered from 28
The chassis should not be ordered from 49
The front should not be ordered from 51
The chassis should not be ordered from 67
Santa's seat should be ordered from 3, 5, or 46
The front should not be ordered from 814
One of the parts should be ordered from 413
The runners should not be ordered from a factor of 17520
Today's clue and the clue on 19 December are false4
The chassis should not be ordered from 52
The chassis should not be ordered from 15
The chassis should not be ordered from 324
Santa's seat should be ordered from 6, 7, or 517
The runners should be ordered from 511
The chassis should not be ordered from 421
One of the parts should be ordered from 83
The front should not be ordered from 119
Exactly 4 of the clues are false15
The chassis should not be ordered from 1, 2, or 823
The chassis should be ordered from 422
The chassis should be ordered from 416
One of the parts should be ordered from 1You 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:
|
|
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 nk 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
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?
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:
1
1
1
0
1
1
1
0
0
1
0
1
0
0
1
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
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.
| – | ÷ | = 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.
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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:
![](javascript:showlimage("advent2025-squares.png"))
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 other
entry 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 the
sum 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
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.
| + | × | = 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?
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