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The Incredible Missing Number Puzzle

I’m thinking of four positive numbers. The numbers can be paired in six possible products. If five of the products are 2, 3, 4, 5, 6, what is the sixth product?

Sources
xkcd forums archived puzzle “four numbers”
https://web.archive.org/web/20190710023801/http://wiki.xkcd.com/irc/Puzzles#Four_Numbers

Russian math forum (sent to me by email)
https://math.d3.ru/neslozhnaia-zadacha-razmiat-mozg-1864141/?sorting=rating

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Video transcription:

Mind Your Decisions, I'm Presh TalwalkarConsider the numbers 1, 2, 3, & 4.There are six possible products that can be made by pairing these numbers. These will be illustrated below.Now to explain the puzzle. I'm thinking of four positive numbers.There are six possible products from pairing the numbers.I will reveal five of the products have values of two, three, four, five and six.What is the value of the sixth product?Pause if you'd like to give this problem a try and when you're ready keep watching to learn how to solve this problem.Before I get to the solution, I'll admit this was a tricky puzzle and I did not solve it myself.But I thought the solution was very clever.So let's go back to our original example of 1, 2, 3 & 4.The trick is to consider the product of 1, 2, 3 & 4 which is equal to 24.Now, let's look at the product from pairing these numbers.Let's exchange these two pairs.Look in the very first line. If we take the product of these pairs, we have 2 times 12.Which is equal to 24.This is also true in the second line,We have 3 times 8 which is equal to 24. And in the very last lineWe have 4 times 6 which is equal to 24. So we found three different ways to get the product of the four original numbers.So suppose the numbers are a, b, c, d and we only know the product of the pairs.So we have four unknown numbers, but we can still make the product a, b, c, d in three different ways.That's the key insight.Furthermore, suppose bc is the unknown pair.Well, if we write that as X, we still have two different ways that we can get the product a, b, c, d. And,the other pair ad which is not listed will pair with the unknown product to getabcd. SoNow, let's solve our problem. We know that the pairs are equal to 2, 3, 4, 5 & 6.So we need to consider,pairing each of these and then examining the product of each of these pairs.We're hoping to get two values that are equal. Because that will be the product of the four original numbers.When we examine these products, we have two ways that we can make 12.So this will reveal that 12 is the product of the four original numbers.We have 2 times 6 2 which is 12, 3 times 4 which is 12. And so 5 is the product which is not involved.Therefore five times x,must be equal to 12. And that six product will be equal to 12 over five or2.4 . And that's our answer.Now for a challenge, what could the four original numbers be?Thanks for making mind your decisions one of the best channels on YouTube. As always, thanks for watching and thanks for your support

Tags:

mathematics,math,maths,riddle,brain teaser,puzzle,math puzzle

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