Sunday Times Teaser 2543 – Hit and Miss

by Victor Bryant

Published: 19 June 2011 (link)

For calculations using arithmetic in a base higher than 10 I need symbols for the higher “digits”. My digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(=10), B(=11), C(=12), etc, using as many letters as necessary. But some numbers are ambiguous. Eg in HIT and in MISS it is unclear whether the second digit is the number 1 or the letter I. So there are four interpretations of HIT + MISS. I’ve taken one of those four interpretations and worked out its remainder when dividing by four. If you knew the remainder you could work out which base I am using.

What is that base?

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Brian Gladman permalink


from itertools import product
from collections import defaultdict

# the values of the letters used
H, I, M, S, T = 17, 18, 22, 28, 29

# a dictionary that maps from the two I values 
# to a map from the (HIT + MISS) mod 4 value
# to the bases that give this value
d = defaultdict(lambda:defaultdict(list))

# the possible base values
for b in range(30, 37):
  # try all four combinations of I values
  for i1, i2 in product((1, 18), repeat=2):
    
    # form HIT and MISS
    hit = (H * b + i1) * b + T
    miss = ((M * b + i2) * b + S) * b + S
    
    # ... and compile the two maps
    d[(i1, i2)][(hit + miss) % 4].append(b)

# scan the maps for a pair of keys that give a 
# unique base value
for rem_to_bases in d.values():
  for l_bases in rem_to_bases.values():
    if len(l_bases) == 1:
      print('The base is {}'.format(*l_bases))

from itertools import product

from collections import defaultdict

# the values of the letters used

H, I, M, S, T = 17, 18, 22, 28, 29

# a dictionary that maps from the two I values

# to a map from the (HIT + MISS) mod 4 value

# to the bases that give this value

d = defaultdict(lambda:defaultdict(list))

# the possible base values

for b in range(30, 37):

# try all four combinations of I values

for i1, i2 in product((1, 18), repeat=2):

# form HIT and MISS

hit = (H * b + i1) * b + T

miss = ((M * b + i2) * b + S) * b + S

# ... and compile the two maps

d[(i1, i2)][(hit + miss) % 4].append(b)

# scan the maps for a pair of keys that give a

# unique base value

for rem_to_bases in d.values():

for l_bases in rem_to_bases.values():

if len(l_bases) == 1:

print('The base is {}'.format(*l_bases))

Sunday Times Teaser 2543 – Hit and Miss

by Victor Bryant

Published: 19 June 2011 (link)

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