-   +   1
+   +   +  
  -   x   -1
-   ÷   -  
  ÷   x   16
4   12   0  

 

in least common multiple by

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1 Answer

Assume initially that all the numbers are different and that, since there are 9 of them, they represent the integers from 1 to 9. Call the unknown known numbers A to I, in order:

A-B+C=1

D-E*F=-1

G/H*I=16

A+D-G=4

B+E/H=12

C+F-I=0

GI=16H; if H=1, (G,I)=(2,8) or (8,2); if H=2 (G,I)=(4,8) or (8,4); if H=3 (G,I)=(6,8) or (8,6) so H is 1, 2 or 3. But I=C+F, so if H=1, (G,C,F)=(2,5,3), (2,3,5) and I=8 since I=2 is not possible. So {A B D E}={4 6 7 9}. But if G=2, A+D=6, and no values fit because there are no numbers in the set summing to 6. Therefore H must be 2 or 3.

If H=2, G=8 and I=4, (C,F)=(1,3) or (3,1) and (A,D)=(5,7) or (7,5) leaving {B E}={6 9). From this we have E=6 and B=9, so that B+E/H=12. What about the other equations? A-B+C=1: since B=9, A+C=10 so A=7 and C=3, leaving F=1 and D=5. So the solution so far gives us (A,B,C,D,E,F,G,H,I)=(7,9,3,5,6,1,8,2,4). Substitute these in the equations:

7-9+3=1 OK

5-6*1=-1 OK

8/2*4=16 OK

7+5-8=4 OK

9+6/2=12 OK

3+1-4=0 OK

So we have a solution! Number matrix is:

7 9 3

5 6 1

8 2 4

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