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You may also be interested in this collection of activities from the STEM Learning website, that complement the NRICH activities above.
Powers and Roots
Sticky Numbers
Age 11 to 14
Challenge Level 
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
Generating Triples
Age 14 to 16
Challenge Level
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Standard Index Form Matching
Age 14 to 16
Challenge Level
Can you match these calculations in Standard Index Form with their answers?
Power Countdown
Age 14 to 16
Challenge Level
In this twist on the well-known Countdown numbers game, use your knowledge of Powers and Roots to make a target.
A Question of Scale
Age 14 to 16
Challenge Level
Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?
You may also be interested in this collection of activities from the STEM Learning website, that complement the NRICH activities above.