MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Competitions 147
Figure 5.29: Miscellaneous #3
Problem 61 (Miscellaneous #3). Consider the sequence 1, 1 1 1 , , · · · , , · · ·
2 3 n
and construct the successive-difference triangle shown in Figure 5.29. Prove
that Pascal’s triangle results if we turn the displayed triangle 60◦ clockwise so
that 1 appears at the apex, disregard minus signs, and divide through every row
by its leading entry. [277, pp. 78-79]
Problem 62 (Curiosa #1). If four equilateral triangles be made the sides of
a square pyramid: find the ratio which its volume has to that of a tetrahedron
made of the same triangles. (Answer: Two.) [82, p. 11]
Problem 63 (Curiosa #2). Given two equal squares of side 2, in different
horizontal planes, having their centers in the same vertical line, and so placed
that the sides of each are parallel to the diagonals of the other, and at such a
distance apart that, by joining neighboring vertices, 8 equilateral triangles are
formed: find the volume of the solid thus enclosed. (Answer: 8 4√ 2( √ 2+1)
.) [82,
3
p. 15]