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.

Show answer

Triangles between squares

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

Show answer & extension

Triangle numbers

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

Show answer & extension


Show me a random puzzle
 Most recent collections 

Advent calendar 2019

Sunday Afternoon Maths LXVII

Coloured weights
Not Roman numerals

Advent calendar 2018

Sunday Afternoon Maths LXVI

Cryptic crossnumber #2

List of all puzzles


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


Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2020