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 (well call these
two cases Sunday and Saturday) separately from only
weekdays being chosen (Weekdays).
The procedure:
- Roll both dice and compute the sum of the top faces.
- If the roll is 1-6 (blue 1, white 6), start over.
- [Sunday case] If the sum is six or less, Sunday is one of the chosen
days.
- Place a marker on Sunday.
- Place a marker on the day corresponding to the blue die. This is
bluesday.
- Place a marker on the day corresponding to the sum. This is sum
day.
- The procedure is finished. The three coins mark the chosen days.
Note that Saturday may be among those chosen here.
- Invert both dice (turn them over to the opposite faces). Bluesday and
sum day change as a result.
- [Saturday case] If the sum is five or less, Saturday is one of
the chosen days and the rest will be weekdays.
- Place a marker on Saturday.
- Place a marker on bluesday.
- Place a marker on sum day.
- The procedure is finished. The three markers indicate the chosen days.
- [Weekdays case] Otherwise, the sum is six or seven. Only
weekdays will be chosen. Well use one die to choose a particular
day, and the sum to determine the two other days.
- 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).
- Place a marker on the weekday that comes next after bluesday (skipping
weekends). The next weekday after Friday is Monday.
- If the sum is odd, place a marker on next weekday after the day marked
in (E.2).
- 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.
- The procedure is finished. The three markers indicate the chosen days.
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