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Show that these two puzzles (mentioned by a couple of readers of my word arithmetic pages) have no solutions:
A A
--- ---
BC)DEF BC)DEF
GH GHI
-- ---
IJ JFirst puzzle:
A
---
BC)DEF
GH
--
IJ
D=1. AxB is one digit, so these are the four possibilities:
A=2234
B=3422
C can be almost anything, but we can use it to deduce G and H, so let's list all possible C's (the x's mark those that don't work):
A=2222 2222 33333 4444
B=3333 4444 22222 2222
C=4789 3789 46789 3789
G=6777 8999 77888 9
H=8468 6468 28147 2
x x x x xx xxxx
For each of these remaining possibilities, J can be a few different numbers, and from those we can deduce F:
A=222 222 222 222 222 333 333
B=333 333 333 444 444 222 222
C=444 888 999 333 888 666 999
G=666 777 777 888 999 777 888
H=888 666 888 666 666 888 777
J=579 459 456 579 357 459 456
F=357 015 234 135 913 237 123
x x xx xx xx xxx xxx
If F is not zero, then E is zero. Also G is greater than E, so E cannot be 9. So we can add E and I to the above chart:
A=x22 2x2 xx2 xx2 xx2 xxx xxx
B=x33 3x3 xx3 xx4 xx4 xxx xxx
C=x44 8x8 xx9 xx3 xx8 xxx xxx
G=x66 7x7 xx7 xx8 xx9 xxx xxx
H=x88 6x6 xx8 xx6 xx6 xxx xxx
J=x79 4x9 xx6 xx9 xx7 xxx xxx
F=x57 0x5 xx4 xx5 xx3 xxx xxx
E=x00 5x0 xx0 xx0 xx0 xxx xxx
I=x95 9x4 xx5 xx7 xx5 xxx xxx
F-H involves a borrow in every case, and so E-1-G=I or E+9-G=I. Looking through the above table, we see that none of the columns allows that. So there are no solutions.
Second puzzle:
A
---
BC)DEF
GHI
---
J
E=0 and H=9. Of the possible permutations of A, B, and C, these are the ones in which AxBC = G9I (we can also deduce D as D=G+1):
A=37788
B=62537
C=48674
G=11325
I=26262
D=22436
xx x
Only two of these work, 7x56=392 (with E=0 and D=4) and 8x74=592 (with E=0 and D=6). The only letters left to be deduced are J and F. In the first one J and F must be 1 and 8 in some order, and neither order works. In the second one J and F must be 1 and 3 in some order, and one of these seems to work: 1+2=3. But then there is no borrow in the right column of the diagram, and a borrow was necessary for E=0 and H=9. So there is no solution.