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Keep sending us YOUR OWN alphanumerics and we'll publish them in collections from time to time. The following two came from Jonathan Gill, St Peter's College, Adelaide, Australia.
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There is a one-to-one correspondence between digits and letters, each letter stands for a single digit and each digit is represented by a single letter. How many different solutions can you find?
Ling Xiang Ning(Allan) form Tao Nan School, Singapore, who solves many of are hardest problems, has sent 7 solutions to CARAVAN and 88 solutions to AUSTRALIAN. Is this all there are? Here is one solution to each.
76 | 968 | |
+86 | +529 | |
---- | ---- | |
162 | 1497 |
Soh Yong Sheng, age 12, also from Tao Nan School, Singapore has sent this solution for.
NRICH | + | STARS | = | MATHS |
17230 | + | 48574 | = | 65804 |
and there are al lot more.
We have the following solutions from Allan Ling (Tao Nan School, Singapore): For the equation
M | A | T | H | |
+ | E | M | A | T |
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I | C | A | L |
T has to be 9 or 0, in order for it to satisfy T+A=A. However if T=0, it is impossible, as H+0 is not L. So T has to be 9.
The following are the possible sums (total 59):
4891 | 5791 | 4791 | 2591 | 3491 | 2491 | 2491 |
+2489 | +2579 | +3479 | +4259 | +2349 | +6249 | +3249 |
7380 | 8370 | 8270 | 6850 | 5840 | 8740 | 5740 |
4391 | 2391 | 3291 | 3291 | 3692 | 4592 | 5092 |
+2439 | +5239 | +5329 | +4329 | +4369 | +3459 | +3509 |
6830 | 7630 | 8620 | 7620 | 8061 | 8051 | 8601 |
4092 | 3092 | 5893 | 1893 | 2793 | 1893 | 2793 |
+3409 | +5309 | +1589 | +4189 | +5279 | +5189 | +1279 |
7501 | 8401 | 7482 | 6082 | 8072 | 7082 | 4072 |
1693 | 4593 | 5493 | 2493 | 1493 | 6093 | 1894 |
+4169 | +1459 | +1549 | +6249 | +7149 | +2609 | +5189 |
5862 | 6052 | 7042 | 8742 | 8642 | 8702 | 7083 |
3794 | 2794 | 6594 | 5294 | 1094 | 1094 | 1094 |
+2379 | +5279 | +1659 | +1529 | +7109 | +6109 | +5109 |
6173 | 8073 | 8253 | 6823 | 8203 | 7203 | 6203 |
4795 | 4795 | 3795 | 2795 | 1695 | 1095 | 1095 |
+3479 | +1479 | +2379 | +3279 | +2169 | +7109 | +6109 |
8274 | 6274 | 6174 | 6074 | 3864 | 8204 | 7204 |
1896 | 2496 | 1296 | 1096 | 1096 | 2197 | 1097 |
+2189 | +1249 | +7129 | +7109 | +3109 | +3219 | +4109 |
4085 | 3745 | 8425 | 8205 | 4205 | 5416 | 5206 |
1097 | 1498 | 3298 | 1298 | 2198 | 4098 | 3098 |
+3109 | +2149 | +1329 | +5129 | +3219 | +2409 | +2309 |
4206 | 3647 | 4627 | 6427 | 5417 | 6507 | 5407 |
2098 | 1098 | 1098 | 1098 | |||
+4209 | +5109 | +4109 | +3109 | |||
6307 | 6207 | 5207 | 4207 | |||
Jonathan also proved that the following alphanumeric does not work, that is it cannot have any solutions. Well done Jonathan.
N | R | I | C | H | |
+ | M | A | T | H | S |
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S | T | A | R | S |
If it was an alphanumerics then H = 0 to satisfy 0 + S = S, but then H cannot be zero, otherwise C + 0 (H) = C and not R. We know that C and R cannot both represent the same number therefore
N | R | I | C | H | |
+ | M | A | T | H | S |
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S | T | A | R | S |
cannot be made into an alphanumeric.