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Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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Nine Colours

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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Two and Two

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Kept Apart

Stage: 3 Short Challenge Level: Challenge Level:1
It is clear that there is a unique way to complete the top three rows, as shown (start in the second square of the third row). Thereafter it is possible to complete the fourth row with R and S alternating and the fifth row QPQPQ.
 
 
 
 

This problem is taken from the UKMT Mathematical Challenges.