The Intelligent Horse

[Counter Striker]

Hi this is the challenging problem for every one who is fascinated by mathematics concerning the real world. Hope you enjoy it!! Here goes.....

...My horse, Harry, is very intelligent. He will do any mathematical problem I set him, providing the answer is 1, 2, 3 or 4. When I set him a problem, he places a hoof on a large grid consisting of unit squares and his answer is the number of squares partly covered by his hoof. Harry has circular hoofs of radius 1/4 unit.

...After many years of collaboration, I suspect that Harry no longer bothers to do the calculations, instead merely placing his hoof on the grid completely at random. I often ask him to divide 4 by 4, but only about 1/4 of his answers are right; I often ask him to add 2 and 2, but disappointingly only about p/16 of his answers are right. Is this consistent with my suspicions?

...I decide to investigate further by setting Harry many problems, the answer to which are 1, 2,3 or 4 with equal frequency. If harry is placing his hoof at random, find the expected value of his answers. The averager of Harry's answers turns out to be 2. Should I get a new horse?


Vocab. : "hoof" is the hard part of the foot of horses, and it is modeled as a circle)