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Here is a $100$ grid with some numbers shaded:

multiples of 5 and 7

What do all the numbers shaded blue have in common?
What do you notice about all the numbers shaded pink?
Can you work out why two of the numbers are shaded in a maroon colour?

Now, here is part of a $100$ square shaded in a different way:

shaded pink: 24, 26, 34, 36, 44, 46 shaded blue: 35

Can you explain the shading this time?

Here are some more parts of the $100$ square, each one shaded according to different rules. Can you work out what the rules are for each?
Is there only one solution each time?

shaded blue: 66 and 77; shaded maroon: 88

shaded blue: 45, shaded maroon: 36 and 54

shaded pink: 16; shaded blue: 27

This problem is featured in Maths Trails - Excel in Problem Solving, one of the books in the Maths Trails series written by members of the NRICH Team and published by Cambridge University Press. For more details about the other books in the series, please see our publications page .