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# Sunday Afternoon Maths LIV

Posted on 2016-07-17

## Hat check

Three logicians, A, B and C, are wearing hats. Each has a strictly positive integer written on it. The number on one of the hats is the sum of the numbers on the other two.
The logicians say:
A: I don't know the number on my hat.
B: The number on my hat is 15.
Which numbers are on hats A and C?

Tags: logic

## Combining multiples

In each of these questions, positive integers should be taken to include 0.
1. What is the largest number that cannot be written in the form $$3a+5b$$, where $$a$$ and $$b$$ are positive integers?
2. What is the largest number that cannot be written in the form $$3a+7b$$, where $$a$$ and $$b$$ are positive integers?
3. What is the largest number that cannot be written in the form $$10a+11b$$, where $$a$$ and $$b$$ are positive integers?
4. Given $$n$$ and $$m$$, what is the largest number that cannot be written in the form $$na+mb$$, where $$a$$ and $$b$$ are positive integers?

## Cross diagonal cover problem

Draw with an $$m\times n$$ rectangle, split into unit squares. Starting in the top left corner, move at 45° across the rectangle. When you reach the side, bounce off. Continue until you reach another corner of the rectangle:
How many squares will be coloured in when the process ends?
If you enjoyed these puzzles, check out Sunday Afternoon Maths LXVII,
puzzles about perfect numbers, or a random puzzle.

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