Problem of the Month (March 2020)

An old puzzle is the pitcher problem. You start with a full 8-ounce pitcher, and empty 3-ounce and 5-ounce pitchers, none of which have any markings on them. By pouring water between pitchers until one pitcher is full or one is empty, the puzzle is to get a pitcher containing exactly 4 ounces of water. There are various solutions, the shortest is 8 → 5 → 3 → 8, 5 → 3, 8 → 5 → 3. The arrows indicate the pours and the blue number indicates which pitcher ends up with 4 ounces.

This month we consider the problem of obtaining a pitcher with 4 ounces of water if only some fraction 0<f≤1 of the water poured makes it into the target pitcher, with the rest 1-f leaking out onto the ground. For example, if f=1/2, then 8 → 5 is a solution, and if f=3/4, then 8 → 3 is a solution. The original problem is for f=1.

Clearly there are an infinite number of values of f that lead to solutions, with f=1 the accumulation point. What rational values of f have solutions? What values of f are roots of rational numbers?

ANSWERS

Bryce Herdt found solutions for (4/5)^1/n and (1/2)^1/n. His solution for (4/5)^1/n is 8 → 5, 8 → 3, (8 → 5 → 8)^n/2, and his solution for (1/2)^1/n is (8 → 5, 8 → 3 → 8, 5 → 8)^n/2.

Known Rational f with Solutions

f	Shortest Solution
1	8 → 5 → 3 → 8, 5 → 3, 8 → 5 → 3
1/2	8 → 5
3/4	8 → 3
4/5	8 → 5, 8 → 3, 5 → 8
5/6	8 → 3 → 8 → 5, 8 → 3, 5 → 3 → 5 → 8
6/7	8 → 3, 8 → 5, 3 → 5, 3 → 8, 5 → 3 → 8
7/8	8 → 3, 8 → 5
8/9	8 → 5, 8 → 3, 5 → 3
7/11	8 → 3 → 5, 8 → 5
9/11	8 → 3 → 5, 8 → 3 → 5, 8 → 5, 8 → 3, 5 → 3 → 8, 5 → 8
10/11	8 → 3 → 5, 8 → 3, 8 → 5
11/12	8 → 3 → 5, 8 → 5, 8 → 3, 5 → 3
12/13	8 → 5, 8 → 3, 5 → 3, 5 → 8
13/14	8 → 3 → 8 → 5 → 3 → 8, 5 → 8 → 5 → 3 → 8
15/16	8 → 3 → 5, 8 → 5, 8 → 3, 5 → 3, 5 → 8
15/17	8 → 3 → 5, 8 → 3, 8 → 5, 3 → 5, 3 → 8, 5 → 3 → 8
15/19	8 → 3 → 5, 8 → 3, 8 → 5, 3 → 5, 3 → 8, 5 → 8
18/19	8 → 3 → 5, 8 → 3, 8 → 5, 3 → 5, 3 → 8, 5 → 3, 5 → 8

Known f with Solutions that are Roots of Rationals

f	Shortest Solution
√1/2	8 → 3 → 5, 8 → 3 → 5
√2/3	8 → 3 → 5, 8 → 3 → 5, 8 → 3, 5 → 8
√3/4	8 → 5 → 3, 5 → 8, 3 → 5, 8 → 3 → 5, 3 → 8, 5 → 8 → 5
√4/5	8 → 5, 8 → 3, 5 → 8 → 5
√5/7	8 → 5 → 3 → 5 → 3 → 5, 8 → 5 → 8
√7/8	8 → 5 → 3, 8 → 5 → 8
√7/9	8 → 3, 8 → 5, 3 → 8, 5 → 3 → 8 → 3 → 8, 5 → 3, 8 → 5
√7/11	8 → 5 → 3 → 5, 8 → 3 → 8, 5 → 8
√10/11	8 → 5 → 3 → 5 → 3, 8 → 5 → 8
√9/13	8 → 3 → 5, 8 → 3 → 5, 8 → 3 → 5 → 8, 3 → 5, 8 → 5
√12/13	8 → 5, 8 → 3, 5 → 8, 3 → 5, 8 → 5, 8 → 3, 5 → 3, 5 → 8
√13/14	8 → 5 → 3 → 5 → 3 → 5 → 3, 8 → 5 → 8

∛1/2	8 → 3, 8 → 5, 3 → 8, 5 → 8 → 5
∛2/3	8 → 3, 8 → 5, 3 → 8, 5 → 3 → 8, 5 → 3, 8 → 5 → 8
∛4/5	8 → 5, 8 → 3, 5 → 8 → 5 → 8
∛7/8	8 → 3, 8 → 5, 3 → 8, 5 → 8 → 3, 8 → 5
∛8/9	8 → 3, 8 → 5, 3 → 8, 5 → 8 → 5, 8 → 3, 5 → 3 → 5 → 3
∛7/11	8 → 3, 8 → 5, 3 → 8, 5 → 8 → 3 → 5 → 8 → 5
∛10/11	8 → 3, 8 → 5, 3 → 8, 5 → 8 → 3 → 5 → 8 → 3, 8 → 5

∜1/2	8 → 3, 8 → 5, 3 → 8, 5 → 8 → 5 → 8
∜2/3	8 → 3, 8 → 5, 3 → 8, 5 → 3 → 8, 5 → 3, 8 → 5 → 8 → 5
∜4/5	8 → 5, 8 → 3, 5 → 8 → 5 → 8 → 5
∜7/8	8 → 3, 8 → 5, 3 → 8, 5 → 8 → 5 → 3, 8 → 5 → 8

⁶√4/5	8 → 5, 8 → 3, 5 → 8 → 5 → 8 → 5 → 8 → 5

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/1/20.