How can we solve these problems withing using brute force?
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More commonly, when problems like this are presented, you get the intermediate products (one digit of the multiplier times the multiplicand) which are added to get the final answer. Then you have many more clues to work with. But for MIX * SIX = SCARCE, for example, the fact that you carry the S means M has to be large. E has to be 1,4, or 9 because it is the last digit of a square. We could have X=8, E=4; X=9, E=1; or X=3, E=9. But I am afraid there will be a bunch of brute force in these.
So you have to calculate left hand side.You will get expression similar to the right hand side. Use fact that equality is true if corresponding coefficients are equal.Applying this condition you should get $5$ equations with $8$ unknown variables.