## Tricky Substitution Equation

Trick brain teasers appear difficult at first, but they have a trick that makes them really easy.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?

### 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

Posted by charlottes-odd on Dec 17, 2013 Another good one eighsse! |

Posted by eighsse on Dec 17, 2013 Thanks charlotte |

Posted by koin_era345 on Dec 20, 2013 I don't know if this should be in trick, but I like it nonetheless. |

Posted by eighsse on Dec 20, 2013 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. |

Posted by dalfamnest on Jul 04, 2014 As a trick, I think it's great - top marks!! Thanks for something new and different. |

Posted by eighsse on Jul 04, 2014 Thanks dalfamnest, glad you enjoyed it |

Posted by Zag24 on Jul 21, 2014 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.) |

Posted by eighsse on Jul 21, 2014 Hah! This is true. Very nice, zag. |

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