## The Blended Game

My PRIMES STEP students invented several variations of Penney’s game. We posted a paper about these new games at math.HO arxiv:2006.13002.

In Penney’s game, Alice selects a string of coin-flip outcomes of length *n*. Then Bob selects another string of outcomes of the same length. For example, Alice chooses HHT, and Bob chooses THH. Then a fair coin is tossed until Alice’s or Bob’s string appears. The player whose string appears first wins. In our example, Bob has a greater probability of winning, namely, 3/4. If the first two flips are HH, then Alice wins; otherwise, Bob wins.

The reader can check that HHT beats HTT with 2 to 1 odds. Thus, the game contains a non-transitive cycle it is famous for: THH beats HHT beats HTT beats TTH beats THH.

I already wrote about the No-Flippancy game that my students invented. It starts with Alice and Bob choosing different strings of tosses of the same length.

However, in the No-Flippancy game, they don’t flip a coin but select a flip outcome deterministically according to the following rule: Let *i* ≤ *n* be the maximal length of a suffix in the sequence of “flips” that coincides with a prefix of the current player’s string. The player then selects the element of their string with index *i* + 1 as the next “flip.” Alice goes first, and whoever’s string appears first in the sequence of choices wins.

My favorite game among the invented games is the **Blended game**, which mixes the No-Flippancy game and Penney’s game.

In the game, they sometimes flip a coin and sometimes don’t. Alice and Bob choose their strings as in Penney’s game and the No-Flippancy game. Before each coin flip, they decide what they want by the rule of the No-Flippancy game above. If they want the same outcome, they get it without flipping a coin. If they want different outcomes, they flip a coin. Whoever’s string appears first in the sequence of `flips’ wins.

For example, suppose Alice selects HHT, and Bob selects THH. Then Alice wants H and Bob wants T, so they flip a coin. If the flip is T, then they both want Hs, and Bob wins. If the first flip is H, they want different things again. I leave it to the reader to see that Bob wins with probability 3/4. For this particular choice of strings, the odds are the same as in Penney’s game, but they are not always the same.

This game has a lot of interesting properties. For example, similar to Penney’s game, it has a non-transitive cycle of choices. Surprisingly, the cycle is of length 6: THH beats HHT beats THT beats HTT beats TTH beast HTH beat THH.

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## Cristóbal Camarero:

I think you have copied the 6-cycle in reverse. You have actually said in the preceding paragraph that Bob with THH beats HHT thrice each four games, right?

10 July 2020, 6:55 am## tanyakh:

Thank you, Cristobal,

You are right, I fixed it.

12 July 2020, 1:34 pm## Divicius:

Hello, big fan of your blog from France here.

23 July 2020, 3:50 pmI teach 11 to 15 years old and this post make me think of something I encountered this year.

My question was : Alice wants to choose randomly an integer from 1 to 10 (both included). She only has one coin. Write a way for her to do it.

It was the first time I asked this question in years of teaching. Few found a valid answer and it was always a variation of four toss and ignore 6 possibilities.

I asked myself, is there a more elegant way but didn’t think of one.

If someone does, I’m going to elaborate on that question this year with various dices.

## tanyakh:

Divicius, You might be able to prove that if you want to guarantee a finite number of throws, the probabilities should be p/q, where q is a power of 2. If you need 1/10, the number of tosses is not guaranteed to be finite.

23 July 2020, 5:13 pm## Yogendra Singh:

I read your blog and i think you have copied the 6-cycle in reversible order. You said in that paragraph Bob with THH beats HHT thrice every four games, ok?

24 August 2020, 12:37 pm