{"id":3339,"date":"2022-05-13T14:57:54","date_gmt":"2022-05-13T19:57:54","guid":{"rendered":"https:\/\/laurentlessard.com\/bookproofs\/?p=3339"},"modified":"2022-05-14T09:02:32","modified_gmt":"2022-05-14T14:02:32","slug":"tetrahedral-dice-game","status":"publish","type":"post","link":"https:\/\/laurentlessard.com\/bookproofs\/tetrahedral-dice-game\/","title":{"rendered":"Tetrahedral dice game"},"content":{"rendered":"

This week’s Riddler Classic<\/a> is a game of four-sided dice:<\/p>\n

\nYou have four fair tetrahedral dice whose four sides are numbered 1 through 4.<\/p>\n

You play a game in which you roll them all and divide them into two groups: those whose values are unique, and those which are duplicates. For example, if you roll a 1, 2, 2 and 4, then the 1 and 4 will go into the “unique” group, while the 2s will go into the “duplicate” group.<\/p>\n

Next, you reroll all the dice in the duplicate pool and sort all the dice again. Continuing the previous example, that would mean you reroll the 2s. If the result happens to be 1 and 3, then the “unique” group will now consist of 3 and 4, while the “duplicate” group will have two 1s.<\/p>\n

You continue rerolling the duplicate pool and sorting all the dice until all the dice are members of the same group. If all four dice are in the “unique” group, you win. If all four are in the “duplicate” group, you lose.<\/p>\n

What is your probability of winning the game?<\/p><\/blockquote>\n

My solution:
\n[Show Solution]<\/a><\/p>\n

\n

We can view this game as a Markov chain<\/a>. At any point in time, we are in a particular state<\/em> of the game, and when we re-roll the dice, we transition to a different state. Although there are many possible states (one for each possible way of rolling the four dice), we can use symmetry arguments to reduce the total states to just four. Here they are:<\/p>\n

    \n
  1. All dice are duplicates. This includes cases like (1,1,1,1) but also (2,2,3,3). If we ever arrive at this position, the game ends and we lose.\n
  2. Three dice are duplicates. For example: (1,1,1,2)\n
  3. Two dice are duplicates. For example: (1,1,2,3)\n
  4. No dice are duplicates. For example: (1,2,3,4). If we ever arrive at this position, the game ends and we win.\n<\/ol>\n

    We can associate each state transition with a probability. For example, suppose we roll (1,1,2,3). We are in the “two-duplicate” state. We must re-roll the two 1’s, and four things could happen:<\/p>\n