Sunday Times Teaser 3195 – Garden Divide

by BRG on 15 December 2023

by Howard Williams

Published Friday December 15 2023 (link)

I have a triangular-shaped garden, the sides of which are an exact number of feet long. To improve its usefulness, I’ve decided to partition it by building a straight fence from one corner to the centre of the opposite side. The length of this fence is exactly 51 feet and the side it attaches to is now 26 feet long each side of the fence.

What, in ascending order, are the lengths of the other two sides of my garden?

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BRG permalink


# lengths: fence: a; half side: b; other sides: c, d;
# angle between fence and side: eta:
#
#    c^2 = a^2 + b^2 - 2.a.b.cos(eta)
#    d^2 = a^2 + b^2 + 2.a.b.cos(eta)
#
#    c^2 + d^2 = 2.(a^2 + b^2)

a, b = 51, 26
for c in range(a - b + 1, a + b):
  d2 = 2 * (a * a + b * b) - c * c
  if (d := round(d2 ** (1/2))) ** 2 == d2 and c < d:
    print(f"{c} and {d} ft.")

# lengths: fence: a; half side: b; other sides: c, d;

# angle between fence and side: eta:

# c^2 = a^2 + b^2 - 2.a.b.cos(eta)

# d^2 = a^2 + b^2 + 2.a.b.cos(eta)

# c^2 + d^2 = 2.(a^2 + b^2)

a, b = 51, 26

for c in range(a - b + 1, a + b):

d2 = 2 * (a * a + b * b) - c * c

if (d := round(d2 ** (1/2))) ** 2 == d2 and c < d:

print(f"{c} and {d} ft.")

Frits permalink

You can also break if c > d or use:

for c in range(a – b + 1, int((a * a + b * b) ** (1/2)) + 1):

Reply

John Z permalink


# Using the cosine rule with angles C1 and C2 such that C2 = Pi - C1
# so Cos(C1) = - Cos(C2)
# Cos(C) = (a^2 + b^2 - c^2)/(2 * a * b)

a = 51
b = 26
sq = tuple(x * x for x in range(a + b))

for c1 in range(a - b, a + b):
  if (c22 := (2 * a * a + 2 * b * b - c1 * c1)) in sq:
    print('Sides are:', c1, sq.index(c22))
    break

# Using the cosine rule with angles C1 and C2 such that C2 = Pi - C1

# so Cos(C1) = - Cos(C2)

# Cos(C) = (a^2 + b^2 - c^2)/(2 * a * b)

a = 51

b = 26

sq = tuple(x * x for x in range(a + b))

for c1 in range(a - b, a + b):

if (c22 := (2 * a * a + 2 * b * b - c1 * c1)) in sq:

print('Sides are:', c1, sq.index(c22))

break

Sunday Times Teaser 3195 – Garden Divide

by Howard Williams

Published Friday December 15 2023 (link)

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