Alphabet Soup

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.