# 2 of 4 cards for 1 die

These are equivalent to using using a deck of 4 cards to emulate a 6-sided die.

A. Assign integer values to   a, b, c, d   so that the 6 possible
pairwise sums   a+b,   a+c,   a+d,   b+c,   b+d,   c+d
are   1, 2, 3, 4, 5, 6   though not necessarily in that order.

B. Same as above but with noninteger values for   a, b, c, d .

C. Assign integer values from the range 0 through 6
to   a, b, c, d   so that the 6 possible pairwise absolute
differences   |a-b|,   |a-c|,   |a-d|,   |b-c|,   |b-d|,   |c-d|
are   1, 2, 3, 4, 5, 6   though not necessarily in that order.
Two solutions.

• Side note: In part C, eliminating the restriction to integers 0 through 6 would allow infinitely many solutions, all added the same amount, as with a + kitchen-sink, b + kitchen-sink, c + kitchen-sink, d + kitchen-sink . – humn Jan 1 '16 at 13:30

## 1 Answer

A. Solution:

0, 1, 2, 4 (hey there binary my old friend)

It should be obvious why this solution works.

B. Solution:

-0.5, 1.5, 2.5, 3.5

This is a simple transformation of the solution to A, and should also be obvious

C. Both solutions:

0, 1, 4, 6

because

|0 - 1| = 1, |4 - 6| = 2, |1 - 4| = 3, |0 - 4| = 4, |1 - 6| = 5, |0 - 6| = 6

and

0, 2, 5, 6

because

|5 - 6| = 1, |0 - 2| = 2, |2 - 5| = 3, |2 - 6| = 4, |0 - 5| = 5, |0 - 6| = 6

• Quick and clean, William Z! These puzzles are distillations of a wonderfully more convoluted 2-of-9-cards-for-2-dice puzzle at ken.duisenberg.com/potw/archive/arch99/990416.html. The goal of that puzzle is to have the 36 pairwise sums of 9 values match the distribution of the 36 possible sums of 2 dice (1 way for sum=2, 2 ways for sum=3, ..., 6 ways for sum=7, 5 ways for sum=8, ..., 1 way for sum=12.) – humn Jan 1 '16 at 13:52
• @human That's interesting, thanks for sharing! These puzzles reminded me of one of Simon Tatham's articles: A Pair Of Dice Which Never Roll 7 – Will Jan 1 '16 at 14:07
• @human How did you know he is William Z? – ghosts_in_the_code Jan 2 '16 at 10:23
• @ghosts_in_the_code I removed outdated info from my SE profile today - and changed my name from the default when I did so. – Will Jan 2 '16 at 11:21