Sunday Times Teaser 2932 – Triangulation

by Andrew Skidmore

Published December 2 2018 (link)

Liam plans to make a set of dominoes. They will be triangular, and one face of each domino will have a number at each corner. The numbers run from 0 up to a maximum digit (9 or less), and the set is to include all possible distinguishable dominoes.

With the maximum digit he has chosen the set would contain a triangular number of dominoes. [A triangular number is one where that number of balls can fit snugly in an equilateral triangle, for example the 15 red balls on a snooker table]

How many dominoes will he need to make?

One Comment Leave one →

Brian Gladman permalink

Either create the set of distinguishable dominoes and find its length:


from itertools import product

# return the triangular number for N (if it exists)
def triangular(N):
  t = 8 * N + 1
  r = round(t ** 0.5)
  return (r - 1) // 2 if r * r == t else None
  
# the maximum digit allowed 
for d in range(2, 10):
  # find the length of the set of distinguishable dominoes
  N = len(set(max((a, b, c), (b, c, a), (c, a, b)) 
            for a, b, c in product(range(d + 1), repeat=3)))
  # is this a triangular number?
  T = triangular(N)
  if T is not None:
    print(f'{N} dominoes (digits: 0..{d}, T = {T})')

from itertools import product

# return the triangular number for N (if it exists)

def triangular(N):

t = 8 * N + 1

r = round(t ** 0.5)

return (r - 1) // 2 if r * r == t else None

# the maximum digit allowed

for d in range(2, 10):

# find the length of the set of distinguishable dominoes

N = len(set(max((a, b, c), (b, c, a), (c, a, b))

for a, b, c in product(range(d + 1), repeat=3)))

# is this a triangular number?

T = triangular(N)

if T is not None:

print(f'{N} dominoes (digits: 0..{d}, T = {T})')

or evaluate the formula giving its length:


# return the triangular number for N (if it exists)
def triangular(N):
  t = 8 * N + 1
  r = round(t ** 0.5)
  return (r - 1) // 2 if r * r == t else None
  
# for maximum digit d, the number of distinguishable dominoes:
#   with three different numbers: (d + 1) * d * (d - 1) // 3
#   with two different numbers:   (d + 1) * d
#   with only one number:         (d + 1)
#
#   total = (d + 1).(d^2 + 2.d + 3) // 3

# the maximum digit allowed 
for d in range(2, 10):
  # the number of distinguishable dominoes
  N = (d + 1) * (d * (d + 2) + 3) // 3
  # is this a triangular number?
  T = triangular(N)
  if T is not None:
    print(f'{N} dominoes (digits: 0..{d}, T = {T})')

# return the triangular number for N (if it exists)

def triangular(N):

t = 8 * N + 1

r = round(t ** 0.5)

return (r - 1) // 2 if r * r == t else None

# for maximum digit d, the number of distinguishable dominoes:

# with three different numbers: (d + 1) * d * (d - 1) // 3

# with two different numbers: (d + 1) * d

# with only one number: (d + 1)

# total = (d + 1).(d^2 + 2.d + 3) // 3

# the maximum digit allowed

for d in range(2, 10):

# the number of distinguishable dominoes

N = (d + 1) * (d * (d + 2) + 3) // 3

# is this a triangular number?

T = triangular(N)

if T is not None:

print(f'{N} dominoes (digits: 0..{d}, T = {T})')

Sunday Times Teaser 2932 – Triangulation

by Andrew Skidmore

Published December 2 2018 (link)

Leave a Reply Cancel reply

Recent Posts

Recent Comments

Archives

Categories

Meta