## Choosing Three Days of the Week Using Two Six-sided Dice

Nov/15/2009

There are 35 ways to choose three items from a set of 7, and there are 36 results from rolling a six-sided die twice. The following procedure will succeed about 97% of the time. The rest of the time it will fail, requiring you to start over.

The procedure uses two six-sided dice of different colors with opposite sides summing to seven(standard dice). We call the two colors blue and white. It also makes use of the diagram below, which can be drawn on a scrap of paper. The days are associated with the numbers 1 through 6 (Sunday is not numbered). Markers will be placed on the diagram to indicate which days are chosen. The procedure considers the cases where weekends are chosen (we’ll call these two cases “Sunday” and “Saturday”) separately from only weekdays being chosen (“Weekdays”).

The procedure:
1. Roll both dice and compute the sum of the top faces.
1. If the roll is 1-6 (blue 1, white 6), start over.
2. [“Sunday” case] If the sum is six or less, Sunday is one of the chosen days.
1. Place a marker on Sunday.
2. Place a marker on the day corresponding to the blue die. This is “bluesday”.
3. Place a marker on the day corresponding to the sum. This is “sum day”.
4. The procedure is finished. The three coins mark the chosen days. Note that Saturday may be among those chosen here.
3. Invert both dice (turn them over to the opposite faces). Bluesday and sum day change as a result.
4. [“Saturday” case] If the sum is five or less, Saturday is one of the chosen days and the rest will be weekdays.
1. Place a marker on Saturday.
2. Place a marker on bluesday.
3. Place a marker on sum day.
4. The procedure is finished. The three markers indicate the chosen days.
5. [“Weekdays” case] Otherwise, the sum is six or seven. Only weekdays will be chosen. We’ll use one die to choose a particular day, and the sum to determine the two other days.
1. Place a marker on bluesday. Note that the blue die cannot be 6 because that would correspond to the start-over roll in (A.1).
2. Place a marker on the weekday that comes next after bluesday (skipping weekends). The next weekday after Friday is Monday.
3. If the sum is odd, place a marker on next weekday after the day marked in (E.2).
4. If the sum is even, skip the day that would have been marked in (E.3), and instead mark the next weekday after that. For example, if bluesday is Thursday, the other days marked will be Friday and Tuesday.
5. The procedure is finished. The three markers indicate the chosen days.

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