Make square number snakes by creating chains of integers where adjacent ones have to sum to a square number. You cannot have any number more than once in the snake. Who can make the longest snake?
Then split your snakes up so that the ends can be placed adjacently so that they sum to a square number. Now you have a square number ring! What is the smallest ring you can make? What is the largest?
Great for getting pupils to learn the square numbers…
Is there a mistake in ring 1:
13+17 = 30 (not square square)
Thanks for spotting this
Corrected
It is easy to construct snakes of arbitary length of course. And for rings you can create rings of size (1),(2),3,4,5 etc.
The only question not obvious to me is what is the longest ring you can make.
The rings are very closely related to a stronger concept I am trying to figure out.
I call it SAS (square-additive-sets). As an example of SAS of size 4 and 5:
[216²,567²,1128²,1848²,5496²][0,1198800, 2695680, 22269520]
That is any element from first set added to any element of second set will give a
square number. From this you can create snakes easy.
Just alternate taking an element from each set and stop when you want. A snake constructed this way allow switching of any elements from the same set and it will still be a snake.
The big question for me about SAS is if there exist a 5*5 SAS or even more, what is the highest n such a n*n SAS exist. So far 3 months brute CPU force has not found anything bigger than a 4*5 SAS.
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There is still a mistake in this bit of snake 1
3-18-17
3+18=21 not a square
18+17=35 not a sqaure