536 Puzzles and Curious Problems

278 Answers

229. BLOCKS AND SQUARES

The smallest number of blocks in each box appears to be 1,344. Surround-

ing the hollow square 342 , the first girl makes her square 502 , the second girl

622 , and the third girl 722 , with her four blocks left over for the corners.

230. FIND THE TRIANGLE

The sides of the triangle are 13, 14, and 15, making 14 the base, the height

12, and the area 84. There is an infinite number of rational triangles composed

of three consecutive numbers, like 3, 4, and 5, and 13, 14, and 15, but there

is no other case in which the height will comply with our conditions.

The triangles having three consecutive numbers for their sides, and having

an integral area, are:

3

13

51

193

723

They are found very simply:

4

14

52

194

724

5

15

53

195

725, etc.

52 = 4 X 14 - 4

194 = 4 X 52 - 14

724 = 4 X 194 - 52,

or generally U" = 4 U,,_l - Un-2, or the general mathematical formula: Find
x, so that 3(x^2 - 1) = a perfect square, where 2x, 2x + 1, 2x - 1 are the
sides of the triangle.

231. COW, GOAT, AND GOOSE

As cow and goat eat ¥.is in a day, cow and goose Y60 in a day, and goat and
goose ~o in a day, we soon find that the cow eats 51160 in a day, the goat 1160
in a day, and the goose 1!J60 in a day. Therefore, together they will eat %60 in
a day, or ¥.io. So they will eat all the grass in the field in 40 days, since there
is no growth of grass in the meantime.