Sunday Times Teaser 3206 – Support Measures

by BRG on March 1, 2024

by Peter Good

Published Friday March 01 2024 (link)

Two buildings on level ground with vertical walls were in need of support so engineers placed a steel girder at the foot of the wall of one building and leaned it against the wall of the other one. They placed a shorter girder at the foot of the wall of the second building and leaned it against the wall of the first one. The two girders were then welded together for strength.

The lengths of the girders, the heights of their tops above the ground, the distance between their tops and the distance between the two buildings were all whole numbers of feet. The weld was less than ten feet above the ground and the shorter girder was a foot longer than the distance between the buildings.

What were the lengths of the two girders?

From → Uncategorized

3 Comments Leave one →

BRG permalink

A very easy one this week.


is_square = lambda x: r if (r := round(x ** (1 / 2))) ** 2 == x else None

maxv = 100

# the gap between the buildings
for w in range(1, maxv):
  # the length of the shorter girder
  p = w + 1
  # the height of the top of the shorter girder [f^2 = p^2 - w^2]
  if not (f := is_square(2 * w + 1)):
    continue
  # the height of the top of the longer girder
  for g in range(f, maxv):
    # the height of the weld [1 / h = 1 / f + 1 / g]
    if (h := g * f / (g + f)) >= 10:
      continue
    # the length of the longer girder [q^2 = w^2 + g^2, q > p]
    if not (q := is_square(w ** 2 + g ** 2)) or q <= p:
      continue
    # the distance between the tops of the girders [v^2 = w^2 + (g - f)^2] 
    if not (v := is_square(w ** 2 + (g - f) ** 2)):
      continue
    print(f"{p = }, {q = }, {f = }, {g = },{ w = }, {v = }, {h = :.3}")

is_square = lambda x: r if (r := round(x ** (1 / 2))) ** 2 == x else None

maxv = 100

# the gap between the buildings

for w in range(1, maxv):

# the length of the shorter girder

p = w + 1

# the height of the top of the shorter girder [f^2 = p^2 - w^2]

if not (f := is_square(2 * w + 1)):

continue

# the height of the top of the longer girder

for g in range(f, maxv):

# the height of the weld [1 / h = 1 / f + 1 / g]

if (h := g * f / (g + f)) >= 10:

continue

# the length of the longer girder [q^2 = w^2 + g^2, q > p]

if not (q := is_square(w ** 2 + g ** 2)) or q <= p:

continue

# the distance between the tops of the girders [v^2 = w^2 + (g - f)^2]

if not (v := is_square(w ** 2 + (g - f) ** 2)):

continue

print(f"{p = }, {q = }, {f = }, {g = },{ w = }, {v = }, {h = :.3}")

Following Frits and Jim Randell, here is a version that eliminates the arbitrary upper limit I use above


from number_theory import is_square, divisor_pairs

# Girder lengths P and Q with P < Q, top heights F and G with F < G
# and weld height H gives 1/H + 1/F + 1/G and a maximum F < 20 when
# F == G. A distance W between buildings and P equal to W + 1 shows
# that W = (F^2 - 1) / 2 with F odd

# the shorter girder height above ground
for F in range(3, 20, 2):
  # the distance between the buildings
  W = (F * F - 1) // 2
  # the shorter girder length
  P = W + 1
  # Since W^2 = Q^2 - G^2 = (Q - G).(Q + G), Q - G is a factor of W^2
  for qmg, qpg in divisor_pairs(W * W):
    # G = (qpg - qmg) / 2 > F
    G, r = divmod(qpg - qmg, 2)
    if r or G <= F:
      continue
    Q = qpg - G
    # the distance between the tops of the girders [v^2 = w^2 + (g - f)^2] 
    if (V := is_square(W ** 2 + (G - F) ** 2)):
      if (H := G * F / (G + F)) < 10:
        print(f"{P = }, {Q = }, {F = }, {G = },{ W = }, {V = } {H = :.2f}")

from number_theory import is_square, divisor_pairs

# Girder lengths P and Q with P < Q, top heights F and G with F < G

# and weld height H gives 1/H + 1/F + 1/G and a maximum F < 20 when

# F == G. A distance W between buildings and P equal to W + 1 shows

# that W = (F^2 - 1) / 2 with F odd

# the shorter girder height above ground

for F in range(3, 20, 2):

# the distance between the buildings

W = (F * F - 1) // 2

# the shorter girder length

P = W + 1

# Since W^2 = Q^2 - G^2 = (Q - G).(Q + G), Q - G is a factor of W^2

for qmg, qpg in divisor_pairs(W * W):

# G = (qpg - qmg) / 2 > F

G, r = divmod(qpg - qmg, 2)

if r or G <= F:

continue

Q = qpg - G

# the distance between the tops of the girders [v^2 = w^2 + (g - f)^2]

if (V := is_square(W ** 2 + (G - F) ** 2)):

if (H := G * F / (G + F)) < 10:

print(f"{P = }, {Q = }, {F = }, {G = },{ W = }, {V = } {H = :.2f}")

John Z permalink


from collections import defaultdict
sq = tuple(i * i for i in range(150))
pythag = defaultdict(list)
pythag1 = dict()
for j in range(1, 99):
  for k in range(j + 1, 100):
    if (c2 := j * j + k * k) in sq:
      h = sq.index(c2) #h: hypotenuse

      if j == h - 1:
        pythag1[j] = (k, h)
      elif k == h - 1:
        pythag1[k] = (j, h)
      else:
        pythag[j].append((k, h))
        pythag[k].append((j, h))

for a, (b, c) in pythag1.items():
  for (e, f) in pythag[a]:
    # a and e - b are two sides of a third right triangle
    for q, r in pythag[e - b]:
      if q == a and (b * a) / (b + e) .lt. 10: # weld height less than 10
        print(a, b, c, 'girder 1:', c )
        print(a, e, f, 'girder 2:', f)
        print(q, e - b, r, 'distance between girder tops:', r)

from collections import defaultdict

sq = tuple(i * i for i in range(150))

pythag = defaultdict(list)

pythag1 = dict()

for j in range(1, 99):

for k in range(j + 1, 100):

if (c2 := j * j + k * k) in sq:

h = sq.index(c2) #h: hypotenuse

if j == h - 1:

pythag1[j] = (k, h)

elif k == h - 1:

pythag1[k] = (j, h)

else:

pythag[j].append((k, h))

pythag[k].append((j, h))

for a, (b, c) in pythag1.items():

for (e, f) in pythag[a]:

# a and e - b are two sides of a third right triangle

for q, r in pythag[e - b]:

if q == a and (b * a) / (b + e) .lt. 10: # weld height less than 10

print(a, b, c, 'girder 1:', c )

print(a, e, f, 'girder 2:', f)

print(q, e - b, r, 'distance between girder tops:', r)

John Z permalink

.lt. in line 23 is supposed to be < .

Reply

Sunday Times Teaser 3206 – Support Measures

by Peter Good

Published Friday March 01 2024 (link)

Leave a Reply Cancel reply

Recent Posts

Recent Comments

Archives

Categories

Meta