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OPEN SQAURES FULL SOLUTION
If we consider about 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 with the term number, for example, for 6th 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 squares’ outer sides and inner sides would always differ by 2, for example, for 6th 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, 6 – 4 = 2.
To prove this:
3rd term: outer side; 3 by 3, inner side; 1 by 1 (total cubes 8, because, 9 – 1 = 8)
4th term: outer side; 4 by 4, inner side; 2 by 2 (total cubes 12, because, 16 – 4 = 12)
5th term: outer side; 5 by 5, inner side; 3 by 3 (total cubes 16, because, 25 – 9 = 16)
6th term: outer side; 6 by 6, inner side; 4 by 4 (total cubes 20, because, 36 – 16 = 20)
7th term: outer side; 7 by 7, inner side; 5 by 5 (total cubes 24, because, 49 – 25 = 24)
8th term: outer side; 8 by 8, inner side; 6 by 6 (total cubes 28, because, 64 – 36 = 28)
9th term: outer side; 9 by 9, inner side; 7 by 7 (total cubes 32, because, 81 – 49 = 32)
10th 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 4 little cubes. After we have done this, we need to find either 2,3, or 4 of these squares that's 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 works. The following are some of the combinations that work:
1,4,12,36
1,4,16,32
1,4,16,36
1,4,20,28
1,4,20,28
1,4,20,32
1,4,24,28
16,12,28
16,10,28
16,10,32
16,36
20,32
20,36
24,28
24,32
…
The rest of the patters should be used with combination equations: nCk = n!/(n-r)!
I think that there are some totals between 50-60 you cannot make, because, some totals need repeated numbers, however, that is not allowed, therefore, some numbers can’t be made, 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 5 different numbers added together.
CHALLENGE 2 SOLUTION
There are some total we cannot make, because, there have to be a pattern, such as, the pyramid goes like this, 1,4,8,12… Therefore, the pyramid have to be layers, it also have a pattern, and if it is once disobeyed, then the total might not be right, and the pyramid might be unbalanced. This is very similar to the last challenge, as it is the same combination, however, it is not limited in numbers perspective, but you can only use 1,2,3 or 4 hollow pyramid to get the total.
We could use the similar idea of the last challenge:
1+4+8+12=25
4+8+12+16=40
8+12+16+20=56
…
I found out you can’t make some total, because you have to do numbers in the patter, such as,14 must be followed by 16, or the shape would not be like a pyramid, it might become a solid cuboid or a hollow cuboid; instead of a pyramid.