The picture above shows a possible initial setup for the game Hey! Thatís My Fish! There are 30 tiles with one fish, 20 tiles with two fish, and 10 tiles with three fish. If the fish are always arranged randomly, but within the same structure of rows (8,7,8,7,8,7,8,7), how many different initial setups are possible? The catch: mirror reflections should not be considered as different.
As we learned in Tuesday's talk, the sequence of Motzkin numbers Mn begins 1, 1, 2, 4, 9, 21, 51, and can be defined as the numbers of ways to get from (0,0) to (n,0), staying on or above the x-axis, taking some combination of the steps (+1,+1) ("U"), (+1,-1) ("D"), and (+1,0) ("L"). Thus for example M3=4, because the paths UDL, ULD, LUD, and LLL all take us from (0,0) to (3,0) without dipping below y=0. We were given a recursion relation for the related sequence of Catalan numbers:
Find (with proof) a similar recursion relation for the Motzkin numbers, and use it to compute the next number in the sequence.
A Pythagorean triple is a triple of positive integers (a,b,c) such that a2+b2=c2. Since the square root of two is irrational, it is impossible for the three lengths a, b, c of a Pythagorean triple to form a 45-45-90 triangle. But, they can come close. I will award some sort of prize for the Pythagorean triple that comes "closest" in some meaningful sense received by Monday, Feb. 4.