# 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

List of all puzzles

## Tags

integration shape division percentages mean algebra regular shapes surds christmas quadratics square numbers odd numbers star numbers functions triangles multiples dodecagons parabolas 2d shapes square roots routes symmetry time sport sum to infinity proportion wordplay area calculus floors crossnumbers ellipses cube numbers angles grids lines people maths clocks money irreducible numbers multiplication coins digits sums perimeter folding tube maps chalkdust crossnumber prime numbers dates graphs rugby differentiation factorials squares planes trigonometry sequences remainders fractions integers arrows factors probabilty scales volume taxicab geometry hexagons spheres palindromes doubling partitions bases polygons logic circles cryptic clues advent games ave speed triangle numbers chess perfect numbers addition menace averages number complex numbers cryptic crossnumbers crosswords probability numbers pascal's triangle shapes chocolate books indices rectangles 3d shapes coordinates dice colouring unit fractions cards balancing means geometry## 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: $$T_n+T_{n+1}+T_{n+2}+T_{n+3}=T_{n+4}+T_{n+5}$$