Sunday Times Teaser 2618 – Antics on the Block

by Andrew Skidmore

Published: 25 November 2012 (link)

A wooden cuboid-shaped block has a square base. All its sides are whole numbers of centimetres long, with the height being the shortest dimension. Crawling along three edges, an ant moves from a top corner of the block to the furthest-away bottom corner. In so doing it travels ten centimetres further than if it had chosen the shortest route over the surface of the block.

What are the dimensions of the block?

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Brian Gladman permalink

Very easy this week – hardly worth a program but here is one:


# Let the cuboid base be b by b and its height h.  With d = 10, 
# we have:
#
#     (2.b + h - d)^2 = b^2 + (b + h)^2
#
#     4.b^2 + h^2 + d^2 + 4.b.h - 2.(2.b + h).d = 2.b^2 + 2.b.h + h^2
#
#     (b - d).(b + h - d) = d^2 // 2
#
# Let (b - d) = f, where f is a factor of d^2 // 2, which gives:
#
#     b = d + f
#     h = d^2 // (2.f) - f
#
# The condition h > 0 means that f < d / sqrt(2). The condition h < b 
# means that f > d / [1 + sqrt(5)]. Hence f is an integer in the range:
#
#     d / (1 + sqrt(5)) < f < d / sqrt(2)
#
# With d = 10, this gives f a factor of 50 in the range 3 < f < 8. So 
# f is 5 giving the cuboid dimensions as 15 x 15 x 5 centimetres.

d = 10
for f in (1, 2, 5, 10, 25, 50):
  b, h = d + f, d * d // (2 * f) - f
  if 0 < h < b:
    print('The cuboid is {0:d} x {0:d} x {1:d} cm'.format(b, h))

# Let the cuboid base be b by b and its height h. With d = 10,

# we have:

# (2.b + h - d)^2 = b^2 + (b + h)^2

# 4.b^2 + h^2 + d^2 + 4.b.h - 2.(2.b + h).d = 2.b^2 + 2.b.h + h^2

# (b - d).(b + h - d) = d^2 // 2

# Let (b - d) = f, where f is a factor of d^2 // 2, which gives:

# b = d + f

# h = d^2 // (2.f) - f

# The condition h > 0 means that f < d / sqrt(2). The condition h < b

# means that f > d / [1 + sqrt(5)]. Hence f is an integer in the range:

# d / (1 + sqrt(5)) < f < d / sqrt(2)

# With d = 10, this gives f a factor of 50 in the range 3 < f < 8. So

# f is 5 giving the cuboid dimensions as 15 x 15 x 5 centimetres.

d = 10

for f in (1, 2, 5, 10, 25, 50):

b, h = d + f, d * d // (2 * f) - f

if 0 < h < b:

print('The cuboid is {0:d} x {0:d} x {1:d} cm'.format(b, h))

Sunday Times Teaser 2618 – Antics on the Block

by Andrew Skidmore

Published: 25 November 2012 (link)

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