Sunday Times Teaser 3006 – Raffle Tickets

by Andrew Skidmore

Published Sunday May 03 2020 (link)

At our local bridge club dinner we were each given a raffle ticket. The tickets were numbered from 1 to 80. There were six people on our table and all our numbers were either prime or could be expressed as the product of non-repeating primes (eg 18 = 2x3x3 is not allowed). In writing down the six numbers you would use each of the digits 0 to 9 once only. If I told you the sum of the six numbers (a perfect power) you should be able to identify the numbers.

List the numbers (in ascending order).

2 Comments Leave one →

Brian Gladman permalink


from number_theory import factor
from itertools import combinations, permutations
from math import exp, log

# the set of numbers in the range 1..80 that are
# primes or the product of non-repeated primes
nbrs = set(x for x in range(2, 81) if x % 11 and
                all(v[1] == 1 for v in factor(x)))

# collect the sequences indexed on their sums
sq2n = dict()
# select the two values less than 10
for a, b in combinations((2, 3, 5, 6, 7), 2):
  
  # select the values for the tens digits of
  # the other four numbers (80 is not valid)
  for nt in range(1, 8):
    t4 = tuple(x for x in range(1, 8) if x not in {a, b, nt})
    
    # use the remaining digits for their units digits    
    for u4 in permutations({0, 8, 9, nt}):
      # form the sequence of six numbers
      n6 = (a, b) + tuple(10 * x + y for x, y in zip(t4, u4))
      # ... all of which are primes or the products of
      # non-repeated primes
      if all(x in nbrs for x in n6):
        
        # save sequences with sums that are perfect powers
        sm = sum(n6)
        for p in range(2, 10):
          v = round(exp(log(sm) / p), 8)
          if v == int(v):
            # save solutions, setting any 
            # duplicate solutions to None
            if sm in sq2n:
              sq2n[sm] = None
            else:
              sq2n[sm] = (n6, int(v), p)

# output solutions with unique sums that
# are perfect powers
for t in sq2n.values():
  if t:
    n6, v, p = t
    print(f"{n6} -> {v ** p} ({v}^{p})")

from number_theory import factor

from itertools import combinations, permutations

from math import exp, log

# the set of numbers in the range 1..80 that are

# primes or the product of non-repeated primes

nbrs = set(x for x in range(2, 81) if x % 11 and

all(v[1] == 1 for v in factor(x)))

# collect the sequences indexed on their sums

sq2n = dict()

# select the two values less than 10

for a, b in combinations((2, 3, 5, 6, 7), 2):

# select the values for the tens digits of

# the other four numbers (80 is not valid)

for nt in range(1, 8):

t4 = tuple(x for x in range(1, 8) if x not in {a, b, nt})

# use the remaining digits for their units digits

for u4 in permutations({0, 8, 9, nt}):

# form the sequence of six numbers

n6 = (a, b) + tuple(10 * x + y for x, y in zip(t4, u4))

# ... all of which are primes or the products of

# non-repeated primes

if all(x in nbrs for x in n6):

# save sequences with sums that are perfect powers

sm = sum(n6)

for p in range(2, 10):

v = round(exp(log(sm) / p), 8)

if v == int(v):

# save solutions, setting any

# duplicate solutions to None

if sm in sq2n:

sq2n[sm] = None

else:

sq2n[sm] = (n6, int(v), p)

# output solutions with unique sums that

# are perfect powers

for t in sq2n.values():

if t:

n6, v, p = t

print(f"{n6} -> {v ** p} ({v}^{p})")

GeoffR permalink

I re-used Brian’s code for generating a set of primes or non-repeated products of primes, then permuted the ten digits to give the six raffle tickets.


from number_theory import factor
from itertools import combinations, permutations
from collections import defaultdict
 
# the set of numbers in the range 1..80 that are
# primes or the product of non-repeated primes
pnrp = set(x for x in range(2, 81) if x % 11 and
                all(v[1] == 1 for v in factor(x)))

# perfect powers to be considered for the sum
pwr_set = {
      2 ** 8,  3 ** 4,  3 ** 5,  5 ** 3,  6 ** 3,  7 ** 3, 
     12 ** 2, 13 ** 2, 14 ** 2, 15 ** 2, 17 ** 2, 18 ** 2
   }

D = defaultdict(set)
# the tens digits are in order so use combinations
for c, e, g, i in combinations(range(1, 8), 4):
  
  # but use a permutation for the remainder
  t = set(range(10)).difference([c, e, g, i])
  for a, b, d, f, h, j in permutations(t):
  
    # form the list of six tickets   
    t = 10 * c + d, 10 * e + f, 10 * g + h, 10 * i + j
    tickets = tuple(sorted((a, b) + t))
    
    # which must sum to a perfect power and all be primes
    # or the product of non-repeating primes
    sm = sum(tickets)
    if sm in pwr_set and all(x in pnrp for x in tickets):
      D[sm] |= {tickets}

# find a set of tickets with a unique sum
for k, v in D.items():
  if len(v) == 1:
    print(f"Tickets: {v.pop()} -> {k}")

from number_theory import factor

from itertools import combinations, permutations

from collections import defaultdict

# the set of numbers in the range 1..80 that are

# primes or the product of non-repeated primes

pnrp = set(x for x in range(2, 81) if x % 11 and

all(v[1] == 1 for v in factor(x)))

# perfect powers to be considered for the sum

pwr_set = {

2 ** 8, 3 ** 4, 3 ** 5, 5 ** 3, 6 ** 3, 7 ** 3,

12 ** 2, 13 ** 2, 14 ** 2, 15 ** 2, 17 ** 2, 18 ** 2

}

D = defaultdict(set)

# the tens digits are in order so use combinations

for c, e, g, i in combinations(range(1, 8), 4):

# but use a permutation for the remainder

t = set(range(10)).difference([c, e, g, i])

for a, b, d, f, h, j in permutations(t):

# form the list of six tickets

t = 10 * c + d, 10 * e + f, 10 * g + h, 10 * i + j

tickets = tuple(sorted((a, b) + t))

# which must sum to a perfect power and all be primes

# or the product of non-repeating primes

sm = sum(tickets)

if sm in pwr_set and all(x in pnrp for x in tickets):

D[sm] |= {tickets}

# find a set of tickets with a unique sum

for k, v in D.items():

if len(v) == 1:

print(f"Tickets: {v.pop()} -> {k}")

Sunday Times Teaser 3006 – Raffle Tickets

by Andrew Skidmore

Published Sunday May 03 2020 (link)

Leave a comment to GeoffR Cancel reply

Recent Posts

Recent Comments

Archives

Categories

Meta