## Choosing Two Numbers from 1..6 Using Two Coin Flips

Nov/15/2009

There are 15 ways to choose three items from a set of 6, and there are 16
results from flipping a two-sided coin four times. The following procedure will
succeed about 94% of the time. The rest of the time it will fail, requiring
you to start over.

Before we start, we divide the numbers into three pairs, {1,4}, {2,5} and
{3,6}. The procedure makes use of the diagram below, which can be drawn on a
scrap of paper. Coins are placed on the diagram to indicate which numbers are
chosen.

The two chosen numbers either fall into the same pair (we’ll call this
“2”) or will fall into two different pairs (“11”). The
procedure will use two flips to decide between these cases, and also to decide
which pairs are non-empty. The final two flips then finish off the selection.

The procedure:
- Flip two coins and place them, in the order they are flipped, on the
numbers 1 and 2.
- Pick up any tails.
- If there is exactly one coin, place a coin on number 3.
- [“11” case] If there are two coins, including step (C) in the
count, they mark the two chosen pairs. We will use one of the numbers from
each of those pairs.
- Consider the chosen pairs from left to right. For each of two flips,
if it is heads, move the marker coin to the higher number in the pair.
Otherwise, leave it on the lower number.
- The procedure is finished. The two coins mark the chosen numbers.

- [“2” case] Otherwise, there are no coins. This indicates that we
have one complete pairs, as yet unchosen.
- Flip two coins and place them, in the order they are flipped, on the
numbers 1 and 2.
- If both are heads, pick them up and start over.
- If no coins are placed, place one on 3.
- Whichever coin is places, place another on the higher number in its
pair.
- The procedure is finished. The two coins mark the chosen numbers.

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