1)You are given 3 disks A,B,C and 3 stands X,Y,Z. The biggest disk is C, then B then A. And A lies on top B which lies on top C all on the same stand, say X. The rule is that the smaller disks must remain on the larger disks.
The following moves will moves these disks together to a new stand.
a)Take A and place it on Y.
b)Take B and place it on Z.
c)Take A and place it on top of B on Z.
d)Take C and place it on Y.
e)Take A and place it on X.
f)Take B and place it on top of C on Y.
g)Take A place it on top of B on top of C on Y.
Now assume you have 64 disks, how long will it take you move them on top a new stand if each move you do is 1 second.
2)Show that if a number is large enough it can be expressed exactly as a sum of 5 postive integral squares.
(Hint: Use 4 Square Theorem)