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Puzzles

25 December

It's nearly Christmas and something terrible has happened: a machine in Santa's toy factory has malfunctioned, and is unable to finish building all the presents that Santa needs. You need to help Santa work out how to fix the broken machine so that he can build the presents and deliver them before Christmas is ruined for everyone.
Inside the broken machine, there were five toy production units (TPUs) installed at sockets labelled A to E. During the malfunction, these TPUs were so heavily damaged that Santa is unable to identify which TPU they were when trying to fix the machine. The company that supplies TPUs builds 10 different units, numbered from 0 to 9. You need to work out which of the 10 TPUs needs to be installed in each of the machine's sockets, so that Santa can fix the machine. It may be that two or more of the TPUs are the same.
You can attempt to fix the machine here.

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24 December

When written in binary, the number 235 is 11101011. This binary representation starts and ends with 1 and does not contain two 0s in a row.
What is the smallest three-digit number whose binary representation starts and ends with 1 and does not contain two 0s in a row?

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23 December

There are 18 ways to split a 3 by 3 square into 3 rectangles whose sides all have integer length:
How many ways are there to split a 10 by 10 square into 3 rectangles whose sides all have integer length?

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22 December

There are 4 ways to pick three vertices of a regular quadrilateral so that they form a right-angled triangle:
In another regular polygon with \(n\) sides, there are 14620 ways to pick three vertices so that they form a right-angled triangle. What is \(n\)?

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21 December

There are 6 two-digit numbers whose digits are all 1, 2, or 3 and whose second digit onwards are all less than or equal to the previous digit:
How many 20-digit numbers are there whose digits are all 1, 2, or 3 and whose second digit onwards are all less than or equal to the previous digit?

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20 December

There are 6 different ways that three balls labelled 1 to 3 can be put into two boxes labelled A and B so that no box is empty:
How many ways can five balls labelled 1 to 5 be put into four boxes labelled A to D so that no box is empty?

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19 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.
+= 7
× × ×
+= 0
÷ ÷ ÷
+= 2
=
4
=
35
=
18

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Tags: numbers, grids

18 December

Some numbers can be written as the product of two or more consecutive integers, for example:
$$6=2\times3$$ $$840=4\times5\times6\times7$$
What is the smallest three-digit number that can be written as the product of two or more consecutive integers?

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