This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

#### Ninety Nine

Every item bought will cause the pence in the total cost to fall by 1. So to spend £65.76, Susanna must have bought

**24**items.##### Extension

What is the smallest amount Susanna could spend for which we could not tell how many items she bought?

#### Reverse Bases

If

*ab*in base 10 is equal to*ba*in base 4, then 10*a*+*b*=4*b*+*a*.So, 9

*a*=3*b*.*a*and

*b*must both be less than 4, as they are digits used in base 4, so

*a*=1 and

*b*=3.

So

**13 in base 10 is equal to 31 in base 4.**By the same method, we find that:

**23 in base 10 is equal to 32 in base 7****46 in base 10 is equal to 64 in base 7**.**12 in base 9 is equal to 21 in base 5.****24 in base 9 is equal to 42 in base 5.**##### Extension

For which pairs of bases

*A*and*B*can you find two digits*g*and*h*such that*g**h*in base*A*is equal to*h**g*in base*B*?
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