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Alphabet Soup

Stage: 3 Challenge Level: Challenge Level:1

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.

C A
+ R A

V A N
A U S
+ T R A

L I A N

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

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

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

S T A R S

cannot be made into an alphanumeric.