Sunday Times Teaser 3064 – Turnip Prize

by BRG on June 11, 2021

by Victor Bryant

Published Sunday June 13 2021 (link)

The Turnip Prize is awarded to the best piece of work by an artist under fifty. This year’s winning entry consisted of a mobile made up of many different plain white rectangular or square tiles hanging from the ceiling. The sides of the tiles were all whole numbers of centimetres up to and including the artist’s age, and there was precisely one tile of each such possible size (where, for example, a 3-by-2 rectangle would be the same as a 2-by-3 rectangle). Last week one of the tiles fell and smashed and then yesterday another tile fell and smashed. However, the average area of the hanging tiles remained the same throughout.

How old is the artist?

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Brian Gladman permalink


def solve():
  ages = []
  nbr, tiles, area = 0, [], 0
  # consider the artist's age
  for age in range(1, 50):
    # accumulate the list of tile areas
    tiles.extend(age * x for x in range(1, age + 1))
    # accumulate the number of tiles 
    nbr += age
    # accumulate the total area of the tiles
    area += age * age * (age + 1) // 2
    # find the (integer) average area
    av_area, r = divmod(area, nbr)
    # look for two or more tiles with this area
    if not r and tiles.count(av_area) >= 2:
      ages.append(age)
  return ages

for age in solve():
  print(f"Age = {age}")

def solve():

ages = []

nbr, tiles, area = 0, [], 0

# consider the artist's age

for age in range(1, 50):

# accumulate the list of tile areas

tiles.extend(age * x for x in range(1, age + 1))

# accumulate the number of tiles

nbr += age

# accumulate the total area of the tiles

area += age * age * (age + 1) // 2

# find the (integer) average area

av_area, r = divmod(area, nbr)

# look for two or more tiles with this area

if not r and tiles.count(av_area) >= 2:

ages.append(age)

return ages

for age in solve():

print(f"Age = {age}")

Runs in 145 microseconds.

Frits permalink


# flatten a list of lists
flatten = lambda group: [x for g in group for x in g]

M = 50        # maximum age + 1

# formula area average: (n + 2) (3n + 1) / 12   (credit: Jim Randell)

# To get a whole number area average,           (credit: Tony Brooke-Taylor)
# n must satisfy the requirement that (n + 2)(3n + 1) mod 12 = 0. 
# This requires that n be equal to either 1 + 12i or 10 + 12i (i=0,1,…).

# generate list of area averages per age
avgs = []
for age in range(1, M):
  if age % 12 not in {1, 10}:
    avgs.append(0)
  else:
    nbr_tiles = age * (age + 1) // 2
    avgs.append(sum([x * x * (x + 1) // 2 for x in range(1, M)][:age]) / 
                     nbr_tiles)

# generate list of areas per age
areas = []
for age in range(1, M):
  if age % 12 not in {1, 10}:
    areas.append([0])
  else:
    areas.append(flatten([[x * y for y in range(1, x + 1)]
                         for x in range(1, M)][:age])) 

ages = []
for i in range(0, M - 1, 3):
  # check for whole number averages which exist more than once in 
  if avgs[i] > 0 and areas[i].count(avgs[i]) > 1:
    ages.append(i + 1)
    
print("age(s):", *ages)

# flatten a list of lists

flatten = lambda group: [x for g in group for x in g]

M = 50 # maximum age + 1

# formula area average: (n + 2) (3n + 1) / 12 (credit: Jim Randell)

# To get a whole number area average, (credit: Tony Brooke-Taylor)

# n must satisfy the requirement that (n + 2)(3n + 1) mod 12 = 0.

# This requires that n be equal to either 1 + 12i or 10 + 12i (i=0,1,…).

# generate list of area averages per age

avgs = []

for age in range(1, M):

if age % 12 not in {1, 10}:

avgs.append(0)

else:

nbr_tiles = age * (age + 1) // 2

avgs.append(sum([x * x * (x + 1) // 2 for x in range(1, M)][:age]) /

nbr_tiles)

# generate list of areas per age

areas = []

for age in range(1, M):

if age % 12 not in {1, 10}:

areas.append([0])

else:

areas.append(flatten([[x * y for y in range(1, x + 1)]

for x in range(1, M)][:age]))

ages = []

for i in range(0, M - 1, 3):

# check for whole number averages which exist more than once in

if avgs[i] > 0 and areas[i].count(avgs[i]) > 1:

ages.append(i + 1)

print("age(s):", *ages)


# formula area average: (n + 2) (3n + 1) / 12   (credit: Jim Randell)
# n = (sqrt(144 * avg + 25) - 7) / 6 is a solution for
# avg(n) = (n + 2) (3n + 1) / 12

from math import ceil

divs = []
# check all possible averages            (avg(49) = 629)
for avg in range(1, 630):
  (age, r) = divmod((144 * avg + 25) ** 0.5 - 7, 6)
  if r:
    continue
  # check divisor pairs of  where both divisors are less equal to 
  for d in range(ceil(avg / age), int(avg ** 0.5) + 1):
    # divisor?
    if not avg % d:
      divs.append([int(age), d])

ages = set()
# check ages for which at least 2 entries exist
for d1 in divs:
   if sum(1 for d2 in divs if d1[0] == d2[0]) > 1:
     ages.add(d1[0])
     
print("age(s):", *ages)

# formula area average: (n + 2) (3n + 1) / 12 (credit: Jim Randell)

# n = (sqrt(144 * avg + 25) - 7) / 6 is a solution for

# avg(n) = (n + 2) (3n + 1) / 12

from math import ceil

divs = []

# check all possible averages (avg(49) = 629)

for avg in range(1, 630):

(age, r) = divmod((144 * avg + 25) ** 0.5 - 7, 6)

if r:

continue

# check divisor pairs of where both divisors are less equal to

for d in range(ceil(avg / age), int(avg ** 0.5) + 1):

# divisor?

if not avg % d:

divs.append([int(age), d])

ages = set()

# check ages for which at least 2 entries exist

for d1 in divs:

if sum(1 for d2 in divs if d1[0] == d2[0]) > 1:

ages.add(d1[0])

print("age(s):", *ages)

Sunday Times Teaser 3064 – Turnip Prize

by Victor Bryant

Published Sunday June 13 2021 (link)

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