### 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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## Comments

Another good one eighsse!

Thanks charlotte

I don't know if this should be in trick, but I like it nonetheless.

It's in Trick because at first sight, the X's will likely be considered letters, which must stand for digits. But they really are multiplication signs, not letters. The only other category I might consider it to be is Situation, as it's somewhat unfair to the solver, considering that the body of the teaser is typed with an actual letter "X" character. So it's written with a third-person perspective since an "X" and a multiplication sign, in writing, are generally interchangeable. This might make it more fit for Situation, but I still feel it is more of a Trick.

As a trick, I think it's great - top marks!! Thanks for something new and different.

Thanks dalfamnest, glad you enjoyed it

Nicely done. Of course, I can beat your solution with my own. No one said that the digits were DECIMAL digits. (Back in the old days, I used to have to add and subtract hexidecimal values all the time. But there's no reason these digits aren't in base 10000000.)

Hah! This is true. Very nice, zag.

AGAIN I DID IT

Too hard & boring.

It's a good teaser but I don't really understand why Karl said Larry's answer was incorrect when the answer was 4

Because he asked what the LARGEST possible value is, not just any valid value.

Solved it its 8

Solved it its 8

Solved it its 8

Good job, Eminem!

Good job, Eminem!

Good job, Eminem!

Good job, Eminem!

Good job, Eminem!

The X's don't have to be multiplication signs to get 8. In algebra, putting variable next to each other assumes that you're multiplying them. This really was just a math question, not a trick.

RiddleTroll, that's very true. I somehow didn't think about that, but I guess the one thing that indicates that these are not algebraic variables is the title. A substitution equation is a particular type of puzzle like this, in which letters are not typical variables, but just placeholders. However, you're still right, because the quotation in the puzzle does not specify that. My mistake!

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