The Hidden Paw

6) Game - Non-Transitive Bet

In the December 1970 issue of Scientific American, Martin Gardner published an article in his “Mathematical Games” column called Nontransitive Paradoxes . (It was also published in his collection Time Travel and Other Mathematical Bewilderments, page 60). He presented a wild collection of paradoxes. One of these had been discovered by a mathematician called Walter Penney. It was published by Penney in the Journal of Recreational Mathematics (October 1969, page 241).

Turning a Paradox into a Bet (or a Game)

Penney’s paradox can be turned into a game. You can use playing cards, colored counters or tossed coins. In the Logical Card Tricks book (Click Here), the victims were named Victor and Victoria. Let us have Victor as the victim this time. He starts by choosing a triplet made up of Reds/Blacks (R/B) or Heads/Tails or the two colors of the counters. You then choose another triplet that is supposed to beat that. As Victor deals the cards or tosses coins or draw counters, he keeps dealing until a triplet comes out that matches Victor's or yours. Whoever gets the first triplet wins that game. Victor will thencontinue dealing, repeating the game as many times as he wishes. In the long run, you will win. (The winning strategy is given below). Here is the procedure for the bet:

1) Victor should select a triplet such as: black-black-black (BBB), black-red-black (BRB), etc.

2) You then announce your triplet (based on the winning strategy below).

3) Victor starts dealing the cards (from a shuffled deck) face up on the table. The first one whose triplet appears will win that run. 

4) Repeat step 3 as many times as Victor wishes. In the long run, you will win.

Use the table below. When Victor selects one of the triplets, locate it in the horizontal row (row 3). You then go down from the triplet in row 3 until you hit a green cell. The triplet in the same row as that green cell (found in col C) would be your triplet. The percentage in the green cell indicates your percentage of the winnings, in the long run.  


For example, if Victor chose the triplet BRB (black red black) in Col F, go down to the green cell in F5 (or combination 1). You must then select BBR which is found in C5. You will win 66.67% of the time. There is another choice (RRB in row 10 or combination 6) that will win, but %62.50 of the time but that is lower than BBR.

Download the Excel Workbook with the Bet

Well, just to assure you that the percentages are correct, I wrote a little computer program in VBA (Excel). Click Here to download the zipped file that contains the workbook. Please write me with your comments (Click Here). 

Winning Strategy

All winning combinations are shown in Green. So how would you remember this when in the pub or after dinner? There is a little mnemonic procedure:

a) Drop the last color from Victor’s triplet. Say he chose BRB, drop the last B. Only BR will remain.

b) Insert a new color before the pair BR and make it the opposite of the second color in the remaining pair. In the case of BR, we reverse the R and insert it before the BR. You will get the BBR triplet (which gets you 66.67%. Another example: for Victor’s choice of RRB, drop the B to get RR. Reverse the second color, R, and insert it before RR to get BRR. The probability of your winning in the long run would be 75%. Not bad!

If this is difficult to remember, switch to binary numbers. Assume that B = 0 and R = 1, the triplets shown in the table become a binary number sequence 000, 001, 010, 011, 100, 101, 110 and 111. So, BRR = 011 which is binary for 3 and so on. (These are shown in the first row above).

To find your winning triplet, remember the number 4163. If Victor chooses 0 or 1, you choose 4 (you can see the two green squares across from 4 or RBB). If he chooses 2 or 3, you choose 1 (the second digit in 4163). And so on.

Click Here for an explanation as to why this procedure works.

Click Here to go to the Weekly Puzzles Summary Page