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# Puzzles

## Archive

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## Elastic Numbers

Throughout this puzzle, expressions like $$AB$$ will represent the digits of a number, not $$A$$ multiplied by $$B$$.
A two-digit number $$AB$$ is called elastic if:
1. $$A$$ and $$B$$ are both non-zero.
2. The numbers $$A0B$$, $$A00B$$, $$A000B$$, ... are all divisible by $$AB$$.
There are three elastic numbers. Can you find them?

## 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

## 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).

## 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.

## 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.

## Largest Odd Factors

Pick a number. Call it $$n$$. Write down all the numbers from $$n+1$$ to $$2n$$ (inclusive). For example, if you picked 7, you would write:
$$8,9,10,11,12,13,14$$
Below each number, write down its largest odd factor. Add these factors up. What is the result? Why?

## 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?

## Subsum

1) In a set of three integers, will there always be two integers whose sum is even?
2) How many integers must there be in a set so that there will always be three integers in the set whose sum is a multiple of 3?
3) How many integers must there be in a set so that there will always be four integers in the set whose sum is even?
4) How many integers must there be in a set so that there will always be three integers in the set whose sum is even?