Sunday Times Teaser 3254 – Pizza Pans

by BRG on 2 February 2025

by Peter Good

Published Sunday February 02 2025 (link)

In a large pan, James baked three identical circular pizzas whose radius was a whole number of cm (less than 75). He laid them on a platter so that one pizza overlapped the other two. The pizza centres formed a right-angled triangle, with sides that were whole numbers of cm. The two lengths of overlap and the gap between the two non-overlapping pizzas (all measured along the lines joining the pizza centres) were all whole numbers of cm and could have formed another right-angled triangle.

He baked a fourth, smaller, circular pizza and it just fitted inside the triangle formed by the centres of the other three. Even if you knew the radii of the pizzas, you couldn’t work out the size of those right-angled triangles.

What was the radius of the smallest pizza?

From → Uncategorized

One Comment Leave one →

BRG permalink


from number_theory import pythagorean_triples
from collections import Counter

# for counting solutions
ctr = Counter()

# consider the radii of the larger pizzas 
for rp in range(1, 75):
  # consider the possible pythagorean triangles formed
  # by the two side overlaps and the hypotenuse gap
  for u, v, w in pythagorean_triples(2 * rp):
    # form the sides of the triangle between the large pizza centres
    a, b, c = 2 * rp - u, 2 * rp - v, 2 * rp + w
    # check that these form a right angled triangle [note c > max(a, b)]
    if a * a + b * b == c * c:
      # find the radius of the smaller pizza
      rs, r = divmod(a * b, a + b + c)
      if not r:
        # count solutions for pairs of pizza radii
        ctr[rp, rs] += 1
        # print(f"{rp = :2} {rs = :2} {f'{(u, v, w)}':>12s} "
        #       f"{f'{tuple(sorted((a, b, c)))}':>16s}")

for (rp, rs), cnt in ctr.items():
  if cnt > 1:
    print(f"Small {rs}cm (Large {rp}cm).")

from number_theory import pythagorean_triples

from collections import Counter

# for counting solutions

ctr = Counter()

# consider the radii of the larger pizzas

for rp in range(1, 75):

# consider the possible pythagorean triangles formed

# by the two side overlaps and the hypotenuse gap

for u, v, w in pythagorean_triples(2 * rp):

# form the sides of the triangle between the large pizza centres

a, b, c = 2 * rp - u, 2 * rp - v, 2 * rp + w

# check that these form a right angled triangle [note c > max(a, b)]

if a * a + b * b == c * c:

# find the radius of the smaller pizza

rs, r = divmod(a * b, a + b + c)

if not r:

# count solutions for pairs of pizza radii

ctr[rp, rs] += 1

# print(f"{rp = :2} {rs = :2} {f'{(u, v, w)}':>12s} "

# f"{f'{tuple(sorted((a, b, c)))}':>16s}")

for (rp, rs), cnt in ctr.items():

if cnt > 1:

print(f"Small {rs}cm (Large {rp}cm).")

Here is a solution based on an analysis (and without imports).


# large pizza radius: r
# side overaps (e, f), hypotenuse gap f: e^2 + f^2 = g^2
# large triangle (pizza radius r):
#     a        b        c      a^2 + b^2 - c^2 == 0 ==> 
#  2.r - e  2.r - f  2.r + g   r = 0 or r = (e + f + g)
#                                   and r = (a + b - c)

rrs_vals = set()
for r in range(1, 75):
  for e in range(1, r // 3):
    for f in range(e + 1, (r - e) // 2 + 1):
      g = r - (e + f)
      if e * e + f * f == g * g:
        a, b, c = 2 * r - e, 2 * r - f, 2 * r + g
        rs, r_ = divmod(a * b, a + b + c)
        if not r_:
          if (rrs := (r, rs)) in rrs_vals:
            print(f"Small {rs}cm (Large {r}cm).")
          rrs_vals.add((r, rs))
          # print(f"{r = :2} {rs = :2} {f'{(e, f, g)}':>12s} "
          #       f"{f'{tuple(sorted((a, b, c)))}':>16s}")

# large pizza radius: r

# side overaps (e, f), hypotenuse gap f: e^2 + f^2 = g^2

# large triangle (pizza radius r):

# a b c a^2 + b^2 - c^2 == 0 ==>

# 2.r - e 2.r - f 2.r + g r = 0 or r = (e + f + g)

# and r = (a + b - c)

rrs_vals = set()

for r in range(1, 75):

for e in range(1, r // 3):

for f in range(e + 1, (r - e) // 2 + 1):

g = r - (e + f)

if e * e + f * f == g * g:

a, b, c = 2 * r - e, 2 * r - f, 2 * r + g

rs, r_ = divmod(a * b, a + b + c)

if not r_:

if (rrs := (r, rs)) in rrs_vals:

print(f"Small {rs}cm (Large {r}cm).")

rrs_vals.add((r, rs))

# print(f"{r = :2} {rs = :2} {f'{(e, f, g)}':>12s} "

# f"{f'{tuple(sorted((a, b, c)))}':>16s}")

Sunday Times Teaser 3254 – Pizza Pans

by Peter Good

Published Sunday February 02 2025 (link)

Leave a Reply Cancel reply

Recent Posts

Recent Comments

Archives

Categories

Meta