Over a decade ago, at the beginning of this blog, I wrote a blog post about some classic math puzzles where a nonsensical result is arrived at by seemingly plausible mathematics; in the post, I challenged folks to figure out the mistake before reading the answers! That post went viral, and was for a while the main driver of traffic to my blog. (Now the main driver is a post about how to hook up a PlayStation 2 to a modern TV.)

This week, I’ve been finishing up my next book project, on the history of invisibility, and in hunting down some references, I had to go to a 1904 issue of The Strand magazine. This led me down a rabbit hole of weirdness, and I wrote a long viral twitter thread about everything I found.

This included another math puzzle, which I post below!

The solution to this puzzle is similar to one from my older post, but I thought I would share it anyway and let people figure out where the math went wrong! A hint, and then the answer, given below the fold.

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HINT: Calculate the totals of 4 – 9/2 and 5 – 9/2 explicitly.

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The mistake is in taking the square root! 4 – 9/2 = -1/2, while 5 – 9/2 = +1/2. The square of both of these quantities is equal to +1/4, but in taking the square root we have kept the negative root on the left side, and the positive root on the right side. We know that there are two possible square roots for any number, but it makes no sense to use the positive root on one side of an equation and the negative root on the other side.

This mistake would be obvious if we had simply written 4 – 9/2 = -1/2 and 5 – 9/2 = +1/2 from the beginning, which goes to show that it is important to make your mathematical formulas as clear as possible if you want to get the correct solution!

Now I need to go back to the 1903 volume of The Strand to see what other mathematical fallacies they published, as implied in the above image…