# Puzzles

## 20 December

18 can be written as the sum of 3 consecutive (strictly) positive integers: 5 + 6 + 7.

18 can also be written as the sum of 4 consecutive (strictly) positive integers: 3 + 4 + 5 + 6.

18 is in fact the smallest number that can be written as the sum of both 3 and 4 consecutive (strictly) positive integers.

Today's number is the smallest number that can be written as the sum of both 12 and 13 consecutive (strictly) positive integers.

## 24 December

There are six 3-digit numbers with the property that the sum of their digits is equal to the product of their digits. Today's number is the largest of these numbers.

## 6 December

Noel's grandchildren were in born in November in consecutive years. Each year for Christmas, Noel gives each of his grandchildren their age in pounds.

Last year, Noel gave his grandchildren a total of £208. How much will he give them in total this year?

## 20 December

What is the largest number that cannot be written in the form \(10a+27b\), where \(a\) and \(b\) are nonnegative integers (ie \(a\) and \(b\) can be 0, 1, 2, 3, ...)?

## Square pairs

Source: Maths Jam

Can you order the integers 1 to 16 so that every pair of adjacent numbers adds to a square number?

For which other numbers \(n\) is it possible to order the integers 1 to \(n\) in such a way?

## 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).

## 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?