||Measuring with unscaled pots — algorithm versus chance
||Szilárd ANDRáS and Örs NAGY
The central focus of this paper is on the following problem: Consider three unscaled pots, with volumes a, b and c ≥ a + b liters, where a, b, c ∈ ℕ*. Initially the third pot is filled with water and the other ones are empty. Characterize all quantities that can be measured using these pots. In the first part of the paper we solve this problem by using the motion of a billiard ball on a special parallelogram shaped table. In the second part we generalize the initial problem for n+1 pots (n ∈ ℕ, n ≥ 2) and we give an algorithmic solution to this problem. This solution is also based on the properties of the orbit of a billiard ball. In the last part we present our observations and conclusions based on a problem solving activity related to this problem. The initial problem for 3 pots is mentioned in  (The three jug problem on page 89), but the solution is not detailed and the general case (with several pots) is not mentioned. The visualization we use is a key element in developing the proof of our results, so the proof can be viewed as a good example of visual thinking used in arithmetic (see , ).