Assuming that none of the numbers contain any leading zero, it can be observed that none of the alphametics: LEVEL+LEVEL = AWESOME or, LEVEL+LEVEL+LEVEL = AWESOME possess any solution. In other words, when LEVEL is added twice (or thrice), it does not equal AWESOME. (Each of the capital letters denotes a different base ten digit from 0 to 9)

(i) What are the respective minimum and maximum number of times that LEVEL can be added to make AWESOME?

(ii) What further solutions are there for (i), assuming that the numbers can contain leading zeroes? Of these additional answers, what are the respective minimum and maximum number of times that LEVEL can be added?

The minimum number of LEVELs added together to equal AWESOME is 37.
AWESOME = 1786508; LEVEL = 48284.

The maximum number of LEVELs added together to equal AWESOME is 575
AWESOME = 9056825; LEVEL = 15751.

An additional five solutions are possible if leading zeroes are permitted, giving a total of twenty-seven solutions. The respective maximum does not change, but the minimum is reduced to two LEVELs:
AWESOME = 0148694; LEVEL = 74347.

0264976 16561
0126972 42324
0148694 74347
0214851 71617
0329712 82428

1047564 24942
1786508 48284
2456905 15851
2658975 35453
2967536 16861
3148054 64246
3541764 24942
4063986 26562
4359025 15851
4726512 32823
5417321 91819
5493789 19691
5728602 32923
6280378 18581
6309120 40704
7053695 15851
7869316 46564
7928102 62426
8074157 97279
8349264 74547
9056825 15751.
9125802 72427

 

Posted by Dej Mar on 2012-09-15 15:30:43