### Brain Teasers

# Tricky Substitution Equation

Karl and his friend Larry are always pulling trick brain teasers on each other. Larry has been getting the best of Karl all too often lately, so Karl really wants to get him back. He comes up with a good one, and writes the following on a piece of paper:

" YXZ

- YXY

= Y"

He shows it to Larry, and says, "Each different letter in this equation stands for a different digit. All instances of a given letter stand for the same digit. There are multiple true solutions, but what is the greatest digit that Y can stand for in a true solution?" Larry scans the equation over and over. He can't come up with any way that the answer could not be 4. "The answer must be four," he says, with confidence. Karl smiles and replies, "Sorry, you're incorrect."

What is the correct answer?

" YXZ

- YXY

= Y"

He shows it to Larry, and says, "Each different letter in this equation stands for a different digit. All instances of a given letter stand for the same digit. There are multiple true solutions, but what is the greatest digit that Y can stand for in a true solution?" Larry scans the equation over and over. He can't come up with any way that the answer could not be 4. "The answer must be four," he says, with confidence. Karl smiles and replies, "Sorry, you're incorrect."

What is the correct answer?

### Hint

The answer is not "Karl is lying. The answer really is 4."And for anyone wondering, the reason 4 is the expected answer is as follows:

The first two digits, "YX," in the first two lines are identical, canceling out to 0, no matter what they stand for. Then the equation resolves to "Z-Y=Y," in which case Z must be 2*Y. This limits Y to a maximum of 4 because Z is a single digit and cannot be greater than 9.

### Answer

The answer is 8. Karl never said that the letters X, Y, and Z are used. Only the letters Y and Z are used in the equation, while the X's are actually multiplication signs. So the teaser actually reads: "Y times Z, minus Y times Y, equals Y," or YZ - Y^2 = YY (Z-Y) = Y

Z-Y = 1

Z = Y+1

So, any digits can be used, as long as they are different and Z is one greater than Y. Z can be 9, with Y being 8, giving "9*8 - 8*8 = 8." This is true, so the answer is valid.

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