My birthday Puzzle, part 1

Computer Science Level 4

508 × S A Y ‾ = H A P P Y ‾ + B I R T H ‾ + D A Y ‾ 508 \times \overline{SAY} = \overline{HAPPY} + \overline{BIRTH} +\overline{DAY}

Each letter represents a different digit from 0 to 9. There are two solutions to this puzzle. State the smaller 9-digit number B I R T H D A Y S ‾ \overline{BIRTHDAYS} .

You may continue to next puzzle here .


The answer is 450928371.

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Chan Lye Lee by Chan Lye Lee , 44, Singapore

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2 solutions

Fletcher Mattox
May 6, 2020

Are you sure there are only two solutions? I found 35. Am I missing something? 

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from itertools import permutationsdigits = range(11)
solution = set()
for (a,b,d,h,i,p,r,s,t,y) in permutations(digits, 10):
    say = 100*s + 10*a + y
    happy = 10000*h + 1000*a + 100*p + 10*p + y
    birth = 10000*b + 1000*i + 100*r + 10*t + h
    day = 100*d + 10*a + y
    if s*h*b*d == 0:
        continue
    if 508*say == happy + birth + day:
        birthdays = int(''.join(list(map(str,[b,i,r,t,h,d,a,y,s]))))
        solution.add(birthdays)
print("smallest solution: %d" % (min(solution)))
print("%d unique solutions" % (len(solution)))
print(sorted(solution))
'''
Output:
smallest solution: 450928371
35 unique solutions
[450928371, 860741592, 1638105402, 1658103402, 1903106452, 3718106902, 3768101902, 4100186732, 4110986532, 4169810532, 5010681732, 50109847
'''

0 Helpful 0 Interesting 0 Brilliant 0 Confused

@Fletcher Mattox , "Each letter represents a different digit from 0 to 9."

Chan Lye Lee - 1 year, 1 month ago
Lu Chee Ket
Nov 30, 2015

The equation is:

50800 S + 4070 A+ 506 Y = 10001 H + 10000 B + 1000 I + 110 P + 100 R + 100 D + 10 T

Generate 1 to 9 for S, H, B and D while 0 to 9 for A, Y, I, P, R and T; filter away any equal among all.

Look for TRUE for the equation introduced above as the conditional statement to quote: 

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S   A   Y   H   B   I   P   R   D   T
1   3   7   2   4   5   6   0   8   9
2   5   9   4   8   6   3   0   1   7

With running figures as stated above, there are i n d e e d indeed two solutions only! The smaller wanted:

508 × \times 137 = 23667 + 45092 + 837 = 69596

BIRTHDAYS = 450928371

LONGINT is chosen using Turbo Pascal.

Answer: 450928371 \boxed{450928371}

0 Helpful 0 Interesting 0 Brilliant 1 Confused

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