# 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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## Tags

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

## Triangles between squares

Prove that there are never more than two triangle numbers between two consecutive square numbers.

## Triangle numbers

Source: ATM Mathematics Teaching 239

Let \(T_n\) be the \(n^\mathrm{th}\) triangle number. Find \(n\) such that: $$T_n+T_{n+1}+T_{n+2}+T_{n+3}=T_{n+4}+T_{n+5}$$