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 was 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, the victims are Victor and Victoria (Click Here). 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. As Victor  deals the cards or tosses coins or draw counters, he keeps doing that until a triplet comes out that matches Victor's or yours. Whoever gets the first triplet wins that game. He will then start all over again, 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 sequence or triplet appears wins that run. 

4) Repeat step 3 as many times as Victor wishes.

According to the table below, after Victor selects one of the triplets in the horizontal row, you should select a triplet that is to the left of the green cell in that column. 


For example, if Victor chose the triplet BRB (black red black), you can choose BBR as the best choice. This means you will win 66.67% of the time. There is another choice (RRB) 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.

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