mscroggs.co.uk
mscroggs.co.uk

subscribe

Puzzles

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?

Show answer & extension

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?

Show answer

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?

Show answer & extension

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?

Show answer & extension

Archive

Show me a random puzzle
 Most recent collections 

Advent calendar 2023

Advent calendar 2022

Advent calendar 2021

Advent calendar 2020


List of all puzzles

Tags

shape dates ave factors books range quadrilaterals integration wordplay functions volume crosswords area squares matrices chocolate square numbers sequences coins elections addition multiples regular shapes indices lines time binary cryptic crossnumbers ellipses probability digital clocks even numbers chalkdust crossnumber palindromes differentiation perfect numbers crossnumbers advent triangles logic graphs sets star numbers pentagons spheres complex numbers dominos parabolas averages trigonometry clocks irreducible numbers number routes proportion axes expansions tangents median circles planes money tournaments determinants percentages digits geometric mean balancing dice arrows grids angles unit fractions remainders rugby polygons scales multiplication means folding tube maps probabilty rectangles decahedra games gerrymandering people maths the only crossnumber geometry symmetry perimeter sport sums floors mean division speed chess factorials sum to infinity combinatorics integers triangle numbers polynomials cube numbers partitions fractions shapes cryptic clues consecutive numbers menace colouring coordinates bases products digital products square roots odd numbers cards dodecagons 2d shapes christmas taxicab geometry prime numbers crossnumber 3d shapes quadratics tiling surds numbers albgebra cubics hexagons pascal's triangle consecutive integers algebra doubling calculus geometric means

Archive

Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2024