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Calcunos

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Here is a solution to Calcunos from Ned who has just left Christ Church Cathedral School in Oxford and is about to go to Abingdon School. Adam of Swavesey Village College, Cambridgeshire also sent in some good work on this investigation.

Dear Bernard

I have solved your question from the July challenges, CalcuNos, as there being 1,374 methods. The numbers of lightbars for each digit are:

1 2 3 4 5 6 7 8 9 0
2 5 5 4 5 6 3 7 6 6

So we only need to consider combinations which add up to 16 using the numbers 2 to 7 and no others.

There are 32 ways of making 16.

7,7,2 6,6,2,2 5,5,4,2 4,4,3,3,2
7,6,3 6,5,5 5,5,3,3 4,4,2,2,2,2
7,5,4 6,5,3,2 5,5,2,2,2 4,3,3,3,3
7,5,2,2 6,4,4,2 5,4,4,3 4,3,3,2,2,2
7,4,3,2 6,4,3,3 5,4,3,2,2 4,2,2,2,2,2,2
7,3,3,3 6,4,2,2,2 5,3,2,2,2,2 3,3,3,3,2,2
7,3,2,2,2 6,3,3,2,2 4,4,4,4 3,3,2,2,2,2,2
6,6,4 6,2,2,2,2,2 4,4,4,2,2 2,2,2,2,2,2,2,2

The number of lightbars is unique except for three numbers which have 5 bars and three which have 6, so it is necessary to work out the number of different ways of arranging each set of numbers and then multiply by three for each of the 5's or 6's involved in the set.

For example:

7, 6, 3: the 3 could go in one of 3 places, the 7 in one of 2 (one has been taken up by the 3) and the 6 in one of 1; this makes 6 combinations.

7,6,3 7,3,6 6,7,3 6,3,7 3,7,6 3,6,7

However, as the 6 can represent any one of three numbers, one must multiply by 3, making a total of 18 combinations for numbers whose digits contain 7, 6 and 3 lightbars.

For combinations like 6,2,2,2,2,2 one sees that, as the 2's must be all the same, only 6 combinations exist (622222, 262222, 226222, 222622, 222262, 222226) times three (for the six), making 18 for this example.

The numbers of combinations for each set of numbers are:

7,7,2 3
7,6,3 18
7,5,4 18
7,5,2,2 36
7,4,3,2 24
7,3,3,3 4
7,3,2,2,2 20
6,6,4 27
6,6,2,2 36
6,5,5 81
6,5,3,2 216
6,4,4,2 36
6,4,3,3 36
6,4,2,2,2 60
6,3,3,2,2 90
6,2,2,2,2,2 18
5,5,4,2 54
5,5,3,3 36
5,5,2,2,2 90
5,4,4,3 36
5,4,3,2,2 180
5,3,2,2,2,2 90
4,4,4,4 1
4,4,4,2,2 10
4,4,3,3,2 30
4,4,2,2,2,2 15
4,3,3,3,3 5
4,3,3,2,2,2 60
4,2,2,2,2,2,2 7
3,3,3,3,2,2 15
3,3,2,2,2,2,2 21
2,2,2,2,2,2,2,2 1

This gives a grand total of 1,374 numbers which, on a calculator, have 16 light bars.