Sunday Times Teaser 3166 – All Their Marbles

by BRG on May 26, 2023

by Peter Good

Published Sunday May 28 2023 (link)

In their art class, Jack and Jill each had a box of spherical glass marbles. Each box contained marbles of at least three different sizes, and the radius of every marble was a whole number of cm. Jack placed three marbles of equal size from his box onto a desk, and placed each of the others in turn on top so that all four marbles touched each other and formed a triangular pyramid. He worked out the height of each pyramid (from desk to top of top marble) and obtained a different whole number of cm each time. Jill was also able to do this with her marbles but they were all of different sizes to Jack’s. None of the pyramids was higher than 30cm.

List all of the different marble radii in ascending order.

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from itertools import combinations, product

# marble radius on table R, on top r, pyramid height h
#
#   h = R + r + sqrt((R + r)^2 - (4/3) * R^2)
#
# with r > (2 / 3^(1/2) - 1).R and 2.R <= h <= 30

# map R to lists of maps of r to h
dR2ldr2h = {}
for R in range(1, 16):
  dr2h = {}
  for r in range(1, 16):
    if r <= (2 / 3 ** (1 / 2) - 1) * R:
      continue
    s2, _r = divmod(3 * (R + r) ** 2 - 4 * R * R, 3)
    if not _r and (s := round(s2 ** (1/2))) ** 2 == s2:
      if 2 * R <= (h := R + r + s) <= 30:
        dr2h[r] = h
  if dr2h:
    dR2ldr2h[R] = dR2ldr2h.get(R, []) + [dr2h]

# collect all possible marble radii
mr = set(dR2ldr2h.keys())
for l_r2h in dR2ldr2h.values():
  mr.update(*(r2h for r2h in l_r2h))

# consider subsets of marble radii of size three or more
d_h2r = {}
for nm in range(3, len (mr) + 1):
  for rds in combinations(mr, nm):
    hts = []
    # collect pyramid heights for all different marble pairings
    for a, b in combinations(rds, 2):
      for a, b in ((a, b), (b, a)):
        if a in dR2ldr2h:
          for dr2h in dR2ldr2h[a]:
            if b in dr2h:
              hts.append(dr2h[b])
    
    # record all marble sets that can make three or more
    # different height pyramids by mapping the pyramid 
    # height groups to the marble sets that make them
    if hts and len(hts) == len(set(hts)) >= 3:
      k = tuple(sorted(hts))
      d_h2r[k] = d_h2r.get(k, set()) | {rds}

p = lambda t, f=2: ' '.join(f"{x:{f}}" for x in t)
# consider pairs of pyramid height groups for Jack and Jill 
for a, b in combinations(d_h2r.keys(), 2):
  # look for two height groups with disjoint marble radii 
  for ar, br in product(d_h2r[a], d_h2r[b]):
    if not (set(ar) & set(br)):
      print(f"Marble radii [{p(sorted(ar + br), 1)}]\n"
            f"  marble radii   ->  pyramids\n"
            f"({p(ar)}) -> ({p(a)})\n({p(br)}) -> ({p(b)})")

from itertools import combinations, product

# marble radius on table R, on top r, pyramid height h

# h = R + r + sqrt((R + r)^2 - (4/3) * R^2)

# with r > (2 / 3^(1/2) - 1).R and 2.R <= h <= 30

# map R to lists of maps of r to h

dR2ldr2h = {}

for R in range(1, 16):

dr2h = {}

for r in range(1, 16):

if r <= (2 / 3 ** (1 / 2) - 1) * R:

continue

s2, _r = divmod(3 * (R + r) ** 2 - 4 * R * R, 3)

if not _r and (s := round(s2 ** (1/2))) ** 2 == s2:

if 2 * R <= (h := R + r + s) <= 30:

dr2h[r] = h

if dr2h:

dR2ldr2h[R] = dR2ldr2h.get(R, []) + [dr2h]

# collect all possible marble radii

mr = set(dR2ldr2h.keys())

for l_r2h in dR2ldr2h.values():

mr.update(*(r2h for r2h in l_r2h))

# consider subsets of marble radii of size three or more

d_h2r = {}

for nm in range(3, len (mr) + 1):

for rds in combinations(mr, nm):

hts = []

# collect pyramid heights for all different marble pairings

for a, b in combinations(rds, 2):

for a, b in ((a, b), (b, a)):

if a in dR2ldr2h:

for dr2h in dR2ldr2h[a]:

if b in dr2h:

hts.append(dr2h[b])

# record all marble sets that can make three or more

# different height pyramids by mapping the pyramid

# height groups to the marble sets that make them

if hts and len(hts) == len(set(hts)) >= 3:

k = tuple(sorted(hts))

d_h2r[k] = d_h2r.get(k, set()) | {rds}

p = lambda t, f=2: ' '.join(f"{x:{f}}" for x in t)

# consider pairs of pyramid height groups for Jack and Jill

for a, b in combinations(d_h2r.keys(), 2):

# look for two height groups with disjoint marble radii

for ar, br in product(d_h2r[a], d_h2r[b]):

if not (set(ar) & set(br)):

print(f"Marble radii [{p(sorted(ar + br), 1)}]\n"

f" marble radii -> pyramids\n"

f"({p(ar)}) -> ({p(a)})\n({p(br)}) -> ({p(b)})")

This is a somewhat surprising teaser since it turns out that there are six combinations of the three base marble radii (R) and the upper marble radii (r) that yield heights but only one of these is a true tetrahedron. In the other five, the tops of all four marbles are at the same height with the resulting ‘tetrahedra’ being no more than flat planes!

Sunday Times Teaser 3166 – All Their Marbles

by Peter Good

Published Sunday May 28 2023 (link)

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