# Puzzles

## Archive

Show me a random puzzle**Most recent collections**

#### Sunday Afternoon Maths LXVII

Coloured weightsNot Roman numerals

#### Advent calendar 2018

#### Sunday Afternoon Maths LXVI

Cryptic crossnumber #2#### Sunday Afternoon Maths LXV

Cryptic crossnumber #1Breaking Chocolate

Square and cube endings

List of all puzzles

## Tags

regular shapes integration area 2d shapes chess star numbers unit fractions integers probabilty volume rugby cryptic crossnumbers logic quadratics means crosswords doubling triangles geometry menace advent money surds lines perfect numbers pascal's triangle ellipses bases sum to infinity cube numbers complex numbers dodecagons mean factors planes differentiation numbers perimeter indices chocolate averages symmetry square roots number arrows folding tube maps shape wordplay algebra graphs spheres shapes functions square numbers prime numbers percentages circles ave speed floors sums routes probability people maths grids triangle numbers addition cards remainders rectangles irreducible numbers partitions dice crossnumbers games coordinates fractions cryptic clues books sequences taxicab geometry factorials 3d shapes calculus scales trigonometry clocks angles christmas squares multiplication sport parabolas odd numbers chalkdust crossnumber multiples dates polygons division digits colouring time balancing hexagons proportion palindromes coins## 15 December

Today's number is smallest three digit palindrome whose digits are all non-zero, and that is not divisible by any of its digits.

## 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, ...)?

## 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:- \(A\) and \(B\) are both non-zero.
- The numbers \(A0B\), \(A00B\), \(A000B\), ... are all divisible by \(AB\).

There are three elastic numbers. Can you find them?

## 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?

## Fill in the digits

Source: Chalkdust

Can you place the digits 1 to 9 in the boxes so that the three digit numbers formed in the top, middle and bottom rows are multiples of 17, 25 and 9 (respectively); and the three digit numbers in the left, middle and right columns are multiples of 11, 16 and 12 (respectively)?

## Always a multiple?

Source: nrich

Take a two digit number. Reverse the digits and add the result to your original number. Your answer is multiple of 11.

Prove that the answer will be a multiple of 11 for any starting number.

Will this work with three digit numbers? Four digit numbers? \(n\) digit numbers?