The deceptive coin

As we have seen in recent weeks, the familiar coin that is tossed in the air to decide something "heads or tails" is equivalent, in the increasingly complex taxonomy of dice, to a two-sided dice, despite that third miniature face. in discord that is the song, and that has elicited interesting comments from readers.

But even though the probability that a normal coin will land on edge is negligible, the two sides of a coin tossed into the air are not equally likely.

The Greek-American mathematician and “mathematician” Persi Diaconis has studied the issue in depth, and has concluded that the face that is facing up at the moment of launch is more likely to come out than the other.

In addition, with the collaboration of his students from Stanford University, Diaconis discovered that most of the coins, if after throwing them in the air we let them fall to the ground, they tend to be more on one side than on the other, because there is a moment they rotate on the edge, which is usually worn unevenly.

This “edge bias” can make the probability that one head will come up up to four times the probability that the other will come up.

If after tossing the coin it is caught between the hands, the bias of the song disappears;

But due to the dynamics of the coin, the face that was up at the time of the launch comes out 51% of the time instead of the expected 50%, as found by Diaconis.

Therefore, the most reliable two-sided dice is not the traditional coin, but the old six-sided dice: instead of playing heads or tails, you play odd or even and the matter settled.

By the way, Diaconis is also an expert in mathematical games with the deck (among other things, he showed that it is enough to shuffle seven times to get a random distribution of the cards), so it will be necessary to return to him some other time.

Rows of coins

And since we have been playing with coins for several weeks, some questions related to them have arisen in the comments section, among others, an interesting problem raised by Salva Fuster:

We have a row of 50 coins of equal or different value (which we could simply substitute for numbers) and there is a game between two players in which each one alternately takes a coin from one end of the row.

The winner is the one with the highest total value with the coins withdrawn.

Is there a strategy that ensures the first player gets a sum equal to at least that of his opponent?

Some reader may have already figured it out at the time of this writing, so don't rush to find the solution in the comments section.

I recommend starting with fewer coins, for example 10, which is a more manageable number and easier to find in any pocket.

There are many and varied problems with coins in a row or in various formations.

Here is a well-known bar counter classic, but worthy of being remembered for its simplicity and elegance: Moving only three coins, get the triangle to invert and rest on its base.

Carlo Frabetti

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