18.000 — Remedial Mathematics Solution

WWWR TOD/RRNDTN ERT RRD ERT TNTT ERT TDNN TTEE TD

From the subtraction TNTT - ERT = TDN, we know T is 1 and N is 0.

From the subtraction TDNN - TTEE = TD, we know D is 2. If we assume base 10 (it is remedial mathematics, after all), then we can also get E = 8 from the ones digits.

From the subtraction RRN - ERT = RR, we know R is 9.

From the last partial product TTEE = R x TOD, knowing TTEE = 1188 and R = 9, we can divide to find TOD = 132, so O = 3.

As noted in the erratum, W does not work. It should be that W x ERT = TOD, but ERT is 891 and TOD is 132, and this requires W = 6.75.

What solvers did: Ignore this problem, and just fill in the letters corresponding to the digits in the given number:

23401340 DO?NTO?N

DOWNTOWN is the answer.

Real solution: Since W x TOD = 4 x 1?2 ends in T (1), and there are seven different letters, the only base which could work is base 7. Following the logic above in base 7, we get E = 5 and R = 6. Then TTEE = 1155₇ = 432, and dividing by R = 6 gives us that TOD is 72 in base 10, or 132₇. Likewise, we now have that W x TOD = ERT with TOD = 132₇ and ERT = 561₇, or 72 and 288 in base 10, so W is 4. Now the whole problem works in base 7:

4446 132/660210 561 662 561 1011 561 1200 1155 12