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From Aardvark to Zebra

A counter-intuitive puzzle

[As heard on Radio 4's Today programme, 9th Feb 2018]

Many of the best puzzles in recreational maths can be traced back to the Victorian puzzlist H.E.Dudeney.

One such puzzle is called The Industrious Bookworm.  A bookworm has chewed through the covers of three volumes of a book sitting on a shelf, and you have to work out how far it has travelled.  The puzzle has a neat twist, but the context is now a bit dated. Modern bookbinding materials are pretty much resistant to boring insects.

So I updated and simplified the puzzle to this:


The Animal Encyclopaedia

I have an encyclopaedia of animals on my shelf, which comes in two volumes.  On the left is the Aardvark to Lynx volume, and next to it on the right is the volume for Mackerel to Zebra.  Each volume is 5cm thick.  The covers are 2mm thick.   I have bookmarked two pages, Aardvark and Zebra.  How far apart would you say the two bookmarks are, to the nearest centimetre?


Before leaping to the solution, it's worth asking yourself how confident you are in your answer.  100% confident?  Or maybe only 90%, because there may be some issue with rounding to the nearest centimetre?  (The puzzle doesn't make it clear if the 5cm thickness of the books includes the covers or not.  This is deliberate.)

I will make a prediction.  If you have never encountered Dudeney's bookworm puzzle, your first answer for the Aardvark puzzle was almost certainly wrong.

Spoiler alert - the solution is below.

                  *                             *                           *                       *

For almost everybody, the debate is about whether the answer is 9cm or 10cm.  But now imagine taking the A-L volume off the shelf. Aardvark is at the start of the first volume, and if you think about it, the first page of the book is on the right of the spine as you look at it on the shelf.  Similarly, Zebra is at the end of the second volume, and the last page is on the left.  So the distance between the two bookmarks is going to be the thickness of the covers plus a couple of pages, 4mm and a bit.  That's 0cm to the nearest centimetre, or maybe 1cm if the book has introduction pages or an appendix.  Certainly not 9cm or more.

This is one of my favourite counter-intuitive puzzles, a familiar everyday situation that unexpectedly catches us out. 


 p.s. If you're wondering what mackerels are doing in an animal encyclopaedia, I couldn't resist a nod to their special place in the puzzle world: