Open Squares

If we consider this, then, each time, except the first term, is a pattern of increasing by four. Or you could say for any term (nth term) the side length of the shape of that term is going to be the same number as the term number, for example, for 6^th term, the side length of that term's shape is going to be 6 by 6, and the hollow part in the middle is going to be 4 by 4, therefore, we can also conclude that any one opens square's outer sides and inner sides would always differ by 2.

For example, for 6^th term, the side length of that term's shape is going to be 6 by 6, and the hollow part in the middle is going to be 4 by 4, therefore, the number of cubes is 6 - 4 = 2.

3^rd term: outer side; 3 by 3, inner side; 1 by 1 (total cubes 8, because, 9 - 1 = 8)

4^th term: outer side; 4 by 4, inner side; 2 by 2 (total cubes 12, because, 16 - 4 = 12)

5^th term: outer side; 5 by 5, inner side; 3 by 3 (total cubes 16, because, 25 - 9 = 16)

6^th term: outer side; 6 by 6, inner side; 4 by 4 (total cubes 20, because, 36 - 16 = 20)

7^th term: outer side; 7 by 7, inner side; 5 by 5 (total cubes 24, because, 49 - 25 = 24)

8^th term: outer side; 8 by 8, inner side; 6 by 6 (total cubes 28, because, 64 - 36 = 28)

9^th term: outer side; 9 by 9, inner side; 7 by 7 (total cubes 32, because, 81 - 49 = 32)

10^th term: outer side; 10 by 10, inner side; 8 by 8 (total cubes 36, because, 100 - 64 = 36)

CHALLENGE 1 SOLUTION

So first what we need to do is find the total of each of the 10 squares, or we can simply find the nth term. For example, the first square would be four little cubes.  After we have done this, we need to find either 2, 3, or 4 of these squares so that the total number of small cubes will add up to a number between 50 and 60.

The pattern is, 1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, however, we only need these numbers: 1, 4, 8, 12, 16, 20, 24, 28, 32, 36 so we can choose any 2, 3 or 4 of these numbers that will add up to a number between 50 and 60, remember, not including 50 or 60.

This would take some time, as you need to work out different combinations that work. The following are some of the combinations that work:

1,4,12,36 = 53

1,4,16,32 = 53

1,4,16,36 = 57

1,4,20,28 = 57

1,4,20,32 = 57

1,4,24,28 = 57

16,12,28 = 56

16,36 = 52

20,32 = 52

20,36 = 56

24,28 = 52

24,32 = 56

I think that there are some totals between 50-60 you cannot make, because some totals need repeated numbers, however, that is not allowed, such as, 59.

Finally, some numbers can't be made, because there could only be 2, 3, 4 numbers used, therefore, you can't use more nor less, however, some numbers require five different numbers added together.

Thank you for your thoughtful response, Frank. I wonder whether anyone has had chance to think about the pyramids in the second challenge?