December 8

Still cutting the paper pieces

 

With 1 cut, - 2 pieces, 2 cuts - 4 pieces, 3 - 7, 4 - 11, 5 - 16...

(Recall triangular numbers: 1, 3, 6, 10, 15,... - do you see any connections?)

Even more cuttings

Can we cut a square with straight cuts? How about pentagon? pentagram?

Of course we cannot cut the pentagram! It is not convex! Only convex figures can be done with straight cuts.

But there is a trick: fold paper! can you cut non-convex stars?

Other triangular numbers

More gallery guards: 

how many you need to guard these galleries?

2, 2, 3 and 2!