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Review of Elastic Numbers

We have a go at the puzzles in Daniel Griller’s new book

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I spend a lot of time solving maths puzzles. Many of my favourites appear in Chalkdust and on my website. But there is a problem with spending so much time doing puzzles: its not very easy for me to find new and interesting puzzles any more.

I was therefore pleased to hear that Daniel Griller—author of the Puzzle Critic blog, a great source of less well-known puzzles including this gem—was releasing a book of original puzzles. Elastic Numbers (Amazon UK, US) is this book, and boasts 108 puzzles. These puzzles are sorted into four sections by difficulty: bronze (easiest), silver, gold and diamond (hardest).

I highly recommend the bronze and silver puzzles to teachers, who will find a collection of well posed questions they can give to students to make them think about common school topics. However, these puzzles don’t offer much challenge to the seasoned puzzler, and although many are neat they feel a little unspectacular.

But the slight disappointment I was feeling about the book immediately disappeared when I flicked forwards to the gold and diamond puzzles. These puzzles will make you immediately reach for the nearest pen and paper and getting solving. With so many good puzzles in these sections, its hard to pick favourites, but the following puzzle stood out (so it’s perhaps not surprising that this puzzle is the source of the title of the book):

Elastic numbers

Source: Elastic Numbers by Daniel Griller (obviously)
A two-digit number $ab$ ($a$ and $b$ are the two digits of the number; the number is not $a$ multiplied by $b$) is called elastic if:

  1. Neither $a$ nor $b$ is zero.
  2. The numbers $a0b$, $a00b$, $a000b$, … made by putting any number of zeros between $a$ and $b$ are all multiples of the original two-digit number $ab$.

Find three elastic numbers, and explain why they are elastic.

As any mathematician will be able to spot, Elastic Numbers is typeset in $\mathrm{\LaTeX}$. I greatly approve of this and the pretty equations it gives (we use $\mathrm\LaTeX$ for Chalkdust too), although this leaves the book looking more like a puzzle dictionary than a fun puzzle book that you might give straight to the kids. But to puzzlers like me, this doesn’t matter: the best thing about a puzzle is the new and exciting mathematical situation it gives you to investigate. And this book is packed full of mathematical excitement. And on that note, I’m off to work out where Evariste is standing…

Where is Evariste?

Source: Elastic Numbers by Daniel Griller (obviously)
Evariste is standing in a rectangular formation, in which everyone is lined up in rows and columns. There are 175 people in all the rows in front of Evariste and 400 in the rows behind him. There are 312 in the columns to his left and 264 in the columns to his right.

In which row and column is Evariste standing?

Matthew Scroggs is a PhD student at UCL working on finite and boundary element methods. His website, mscroggs.co.uk, is full of maths and now features a video of him completing a level of Pac-Man optimally.
Twitter  @mscroggs    Website  mscroggs.co.uk    + More articles by Matthew

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