A curious puzzle by Stanley Rabinowitz, from the Spring 1984 issue of Pi Mu Epsilon Journal:
In the little hamlet of Abacinia, two different base systems are used, and everyone speaks the truth. One resident said, “26 people use my base, base 10, and only 22 people speak base 14.” Another said, “Of the 25 residents, 13 are bilingual and 1 is illiterate.” How many people live in Abacinia?
|
SelectClick for Answer |
This solution is by Rogm Kuehl:
Let the first resident speak base b. Then the second resident speaks base b + 4 since in that base the total population will be represented by a smaller numeral (25) than the numeral used by the first speaker as is the case. The total population is therefore 2(b + 4) + 5 = 2b + 13 . The number of people speaking base b, according to the first speaker, is 2b + 6 and the number speaking base b + 4 is 2b + 2. According to the second speaker 1(b + 4) + 3 = b + 7 people speak both bases and 1 is illiterate. Therefore the total population is
(2b + 6) + (2b + 2) + 1 – (b + 7) = 3b + 2.
Equating this to 2b + 13, we get that the two bases are
b = 11
b + 4 = 15.
Now the total population, 2b + IS, is 35 (base ten).
|
