Sunday Times Teaser 3336 – Save Gardener’s Canes

by BRG on 30 August 2026

by Phil Jackson

Published Sunday August 30 2026 (link)

A gardener had two pairs of bamboo canes, one pair 70 inches long and one pair 94 inches long. Well, he had until his wife ran over them in her car and broke both of the 94-inch canes into 2 pieces. Remarkably all of the broken pieces ended up a whole number of inches long. Even more remarkably when he connected the longest and shortest pieces to one of the 70-inch canes and the two medium pieces to the other 70-inch cane, he was able to make two triangles with enclosed areas a whole number of square inches. He worked out that the enclosed areas differed by exactly 504 square inches.

What was the length of the shortest broken piece?

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BRG permalink

It looks like the bamboo length of 94 inches should be 98.

# the two bamboo lengths

ls, lx = 70, 98

# the maximum length of the first long cane split

for l1x in range(lx // 2, lx + 1):

# the maximum length of the first long cane split (longer)

for l2x in range(l1x + 1, lx + 1):

# the two triangles

t1 = (ls, l2x, lx - l2x)

t2 = (ls, l1x, lx - l1x)

# find the two areas if possible

areas = []

for a, b, c in (t1, t2):

s = (a + b + c) / 2

asq = (s * (s - a) * (s - b) * (s - c))

if asq <= 0:

continue

area = asq ** (1 / 2)

if abs(area - round(area)) < 1e-15:

areas.append(round(area))

# find an area difference of 504 square inches

if len(areas) == 2 and abs(areas[0] - areas[1]) == 504:

print(f"{lx - l2x}, {(lx - l1x, l1x, l2x)}")

John Z. permalink

This one had me stumped until I saw Brian’s 98 -> 94 correction

from itertools import combinations

sol = dict()

# semi perimeter of both triangles

s = (70 + 98) // 2

# consider possible sides x and y (which must sum to 98)

for x in range(1, 50):

y = 98 - x

# save triangles with integral area indexed by short side

if (a2 := s * (s - 70) * (s - x) * (s - y)) > 0:

if (a := a2 ** 0.5) == int(a):

sol[x] = a

# find two triangles that meet difference criterion

for (d, e), (f, g) in combinations(sol.items(), 2):

if g - e == 504:

print("shortest side:", d)

Sunday Times Teaser 3336 – Save Gardener’s Canes

by Phil Jackson

Published Sunday August 30 2026 (link)

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