In the game of Turning Turtles, some coins are placed in a line on a table, from

Question

In the game of Turning Turtles, some coins are placed in a line on a table, from left to right. Each coin has two sides, called heads (H) and tails (T). The coins are placed on the table so that one side is visible, and the other side is facing downwards. We describe a position by writing down for each coin which side is visible (either 'H' or 'T'). For example, "THTTT" denotes the position where there are 5 coins on the table, the leftmost coin is showing the "T" side, the next coin is showing 'H' and the three coins to the right of this are showing "T". A move consists of: flipping exactly one 'H' into a "T", and then (if the player wishes) flipping any coin to the left of this (this second flip can either be from 'H' to 'T' or from 'T' to 'H'). The terminal positions in the game are the positions where all coins are showing 'T'. (a) List all moves from the position "HTHTT". (b) Find the P-positions among all positions where there is exactly one "H" (and all remaining coins are showing "T"). (c) Identify all P-positions.

Explanation / Answer

a) HTHTT

   If he flips the 1st H, then TTHTT and move ends.

   In next move, he flips the other H to obtain TTTTT and it's a terminal position.

   Player can flip the H in position 3 first for HTTTT and stop or further flip the T in 2nd to H to get HHTTT.

   Again, a move could be THTTT, then TTTTT or HTTTT and TTTTT

   So, all moves are {TTHTT, HTTTT, HHTTT, THTTT, TTTTT}

b) For p-positions with exactly one 'H', we have p-1 possibilities starting from position 1,2,3 ... p-1

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