Find the 6-digit number with no zeros where:
- No digit appears more than once.
- The sum of the last four digits equals twice the sum of the first two.
- The first two digits form a prime number when concatenated.
- Digits 2 and 3 also form a prime.
- Digits 3 and 4 also form a prime.
- Digits 4 and 5 also form a prime.
- Digit 5 itself is also a prime.
- All six digits add up to 33.
The last two digits need not form a prime.
From Mensa Puzzle Calendar 2018 by Fraser Simpson, Workman Publishing, New York. Puzzle for October 9.
831975
reasoning:
with sum 33 digits 1 & 2 add to 11. Only prime possibilities are
29, 47, and 83. So we run through all chains of possible primes in ascending order:
29 297 2971 29713 (last 4 can't add to 22)
29 297 2973 29731 (last 4 can't add to 22)
47 471 4713 (no more primes)
47 471 4719 (no more primes)
47 473 4731 47319 (9 not prime)
47 479 (no more primes)
83 831 8317 83179 (9 not prime)
83 831 8319 83197 831975 (works)
83 837 8371 83719 (9 not prime)
83 837 8379 (no more)
QED
(BTW, if it is a Mensa Calendar, why did they need to add "The last two digits need not form a prime? :-) )
p&p soln
