I get irritated when somebody multiplies two numbers and gets the wrong answer, and then says, “I’m bad at math.”  That’s an overstatement that gives them undeserved status. They are bad at arithmetic, which is a very specialized area of math. Adding two numbers incorrectly and saying, “I’m bad at math” is an insult to mathematicians everywhere.

With this in mind, let’s talk about the arithmetic in Fermat’s Last Theorem. Suppose I have some number x and multiply it by itself n times. n is an arbitrary number. It can be 2, 3, 7, 10, or 22. We then choose another number and also multiply it by itself n times. Let’s call this second number y. We do the same thing for a third number z.

Fermat’s theorem in mathematics says that you cannot find an x and a y, multiply each number by itself n times, and have it equal to a third number z multiplied by itself n times. It only works for n is equal to 2, but it does not work for n larger than 2. In terms of math, this can be summarized as

All of the numbers used here must be positive whole numbers.

 Pierre de Fermat (1601‒1665) first stated the theorem in 1637 in the margin of his copy of Arithmetica, an ancient book of algebra problems and solutions. For 358 years, Fermat’s Last Theorem, as it came to be called, remained an open problem but Andrew Wiles proved it in 1994‒1995.

A friend of mine in mathematics, Dave Johns, texted me to say that he believed Wiles’ proof was incorrect. He had found a counter example. Specifically, he texted three numbers that he said disproved the famous Theorem:

x = 321,232,430,786,123,860

y = 487,367,975,876,123,125

 z = 4,534,432,154,871,543,981

The text ended there. He never told me what n was. Dave later had a mild stroke that destroyed part of his memory so the identity of n remained a mystery.

I recently showed the numbers to another friend, Pythagoras Smith.  (His mom  named all her children after mathematicians.) My friend, who we affectionately call Pithy, got his PhD in math when he was 14. He looked at the numbers for 30 seconds and, without a computer or performing any multiplication, announced this could not be a solution for any n. Even n = 2.

This week’s Micro Softy is a rough one.  How could Pithy know this so quickly?

Hint: Consider the title of this article.

Solution to Micro Softy 79:  Card Sharks That Bite Harder

Here’s the substance of  last week’s Micro Softy: Two identical decks of standard playing cards (52 cards each, no jokers) are combined and thoroughly shuffled, giving a total of 104 cards. The shuffled stack is then split into two piles of 52 cards each. The puzzle comes in two parts.

Part 1: In the left pile, count the number of red cards (hearts and diamonds). In the right pile, count the number of black cards (spades and clubs). What is the probability that these two counts differ by no more than 2?

Part 2: Now suppose each deck originally included two jokers, making 54 cards per deck. After combining and shuffling both decks (108 cards total), the cards are again split into two piles of 54. What is the probability in this case that the number of red cards in the left pile differs from the number of black cards in the right pile by at most 2?

Here’s the answer: In both cases, the number of red cards in the left piles is exactly equal to the number of black cards in the other. Always.

Posing the problem as a probability was a distraction. The final answer is that, in both cases, the probability that the number of cards is equal is one (1).

The Monday Micro Softy is a weekly feature of Mind Matters News. Here are the links to all the puzzles and answers to date:

Monday Microsofty 78: Card Sharks That Bite Harder. You can beat the odds in some card games if you understand probability theory. Try your chances! Last week’s puzzle, like several others, is easy to solve if we use inclusive thinking about relationships.

Monday Micro Softy 77: Two Proud Texans  I’m aware of no other state where businesses and citizens proudly fly their state flag. I live in McGregor, Texas, where Elon Musk’s Space X has a testing center, and occasionally, the testing of their rocket engines gently rattles the dishes on the shelves in my home. 

Monday Micro Softy 76: The Smoking Gun explains a computer scientist or engineer, a law enforcement officer often relies on abductive reasoning to crack a case, so with this in mind, you will have to crack last week’s puzzle. You can find puzzles 55 through 75 here as well.

Monday Micro Softy 55: “It happens every spring.” Baseball, that is. Here’s a puzzle that takes in baseball’s summer. To solve last week’s puzzle, you don’t need to know the distance. Check the problem again for the number you do need to know. You can find puzzles 51 through 54 here as well.

Monday Micro Softy 50: Cutting through the cornbread. How did Yuri Senior cut the cornbread into eight identical portions using only three straight cuts? You can guess the answer to Microsofty 49 if you try the test question yourself at home, using a small mirror. Links to Microsofties 46 through 49 are here as well.

Monday Micro Softy 45: Can Tony beat the fast-food curfew? An early curfew on fast food service motivated a boy to exercise more vigorously. But how fast was he pedalling? To solve Micro Softy 44, recall that Tony doesn’t need to take the individual pills each day, only the prescribed amount of each. You will find links here to Micros Softies 41 through 44 as well.

Monday Micro Softy 40: The fate of a false prophet. He wasn’t actually fired for being a false prophet but for something that his prophecy unintentionally revealed. The solution to Micro Softy 39 lies in considering an alternative possible meaning of a word commonly used in sports. You will also find links to Micro Softies 30 through 39 and their answers here as well.

Monday Micro Softy 29: A funeral lament in four lines. The funeral director was puzzled by Dan’s description of his relationship to the deceased but there was no question that his grief was sincere Here, you will also find links to Microsofties 22 through 29.

Monday Micro Softy 21: Finding More of the Deadly Fentanyl Pills. Here, you will also find links to Microsofties 11 through 20 as well.

Monday Micro Softy 11: What Happened to That Other Dollar? Here you will find links to the first ten Micro Softies. Have fun!