Solution to Princess Gwendolynn’s Puzzle:
Make a chart of the answers; note where patterns begin & end (powers of 2):
20
21
22
23









skip 1, delete 2
skip 1, delete 2, skip 3; delete 1
sk1, dl2, sk3, dl4; sk1, dl3
sk1, dl2, sk3, dl4, sk5; dl1
sk1, dl2, sk3, dl4, sk5, dl6; sk1, dl3, sk5; dl3
etc…
(notice that all even numbers are eliminated
on the first round)
x = # of knights
c = chair of choice
So we have:
2(x - 2n) + 1 = c
and
2(n+1) > x  2n
because we are counting every other knight
because we skipped the first chair (1)
because we have no negative or zero chairs, and we are only
concerned with the remainder beyond a power of two
For 20 chairs:
x = 20
2n = 24 = 16
2(20 – 16) + 1 = 2(4) + 1 = 9
For 7 chairs:
x=7
2n = 22 = 4
2(7 – 4) + 1 = 2(3) + 1 = 7
For 1 chair:
x=1
2n = 2 0 = 1
2(1 – 1) + 1 = 2(0) + 1 = 1
For 2 chairs:
x=2
2n = 21 = 2
2(2 – 2) + 1 = 2(0) + 1 = 1