The identity also admits a natural probabilistic interpretation as follows. Let be four integer numbers independently and uniformly chosen at random between and . Then, the probability that is the largest of the four numbers equals the probability that is at least as large as and that is at least as large as . That is, formula_8. For any particular value of , the combinations of , , and that make largest form a cube so (adding the size of this cube over all choices of ) the number of combinations of for which is largest is a sum of cubes, the left hand side of the Nichomachus identity. The sets of pairs with and of pairs with form isosceles right triangles, and the set counted by the right hand side of the equation of probabilities is the Cartesian product of these two triangles, so its size is the square of a triangular number on the right hand side of the Nichomachus identity. The probabilities themselves are respectively the left and right sides of the Nichomacus identity, normalized to make probabilities by dividing both sides by .