# Puzzles

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#### 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

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

Source: UKMT 2011 Senior Kangaroo

How many positive two-digit numbers are there whose square and cube both end in the same digit?

## What's the star?

In the Christmas tree below, the rectangle, baubles, and the star at the top each contain a number. The square baubles contain square numbers; the triangle baubles contain triangle numbers; and the cube bauble contains a cube number.

The numbers in the rectangles (and the star) are equal to the sum of the numbers below them. For example, if the following numbers are filled in:

then you can deduce the following:

What is the number in the star at the top of this tree?

*You can download a printable pdf of this puzzle here.*