Three-E-Mail Puzzles

By Scott, Paul | Australian Mathematics Teacher, Summer 2011 | Go to article overview

Three-E-Mail Puzzles


Scott, Paul, Australian Mathematics Teacher


I receive e-mails from many people, but from time to time I get sent a mathematical offering. This is usually from people who know of my interest in such things, and who want an answer! These puzzles are usually nicely dressed up, and fun to solve. See what you think!

Puzzle 1

A little something to wake up your brain:

What is the sum in the second row?

  [        [        [        o       = 11
  o        Y        o        U       = ?
  U        Y        U        U       = 11
  o        o        o        [       = 9
 = 8      = 21     = 8      = 7

This is a mathematical challenge, and it's been said that:

* if you're an engineer, you should be able to solve it in (under) three minutes;

* if you're an architect, in three hours;

* if you're a doctor, in six hours;

* if you're an accountant, in three months and

* If you're a lawyer, probably never...

What is your vocation?

We might see this as a simple problem in linear algebra, especially if we replace the weird symbols by the more familiar x, y, z and w, obtaining

  x        x        x        y       = 11
  y        z        y        w       = ?
  w        z        w        w       = 11
  y        y        y        x       = 9
 = 8      = 21     = 8      = 7

From row 1 we find 3x + y = 11. Assuming that our unknowns are positive integers, this means that either x = 1, y = 8 or x = 2, y = 5 or x = 3, y = 2.

From row 4, we have 3y + x = 9, so the only possibility is that x = 3 and y = 2.

Column 1 now tells us that w = 1, and column 2 that z = 8.

We deduce that the missing number is 13.

This obviously makes me an engineering candidate, but I am not happy. Good problems usually supply a minimum of information, and this problem has a superabundance of (perhaps redundant) information that I have not used. Perhaps I have missed something here?

Puzzle 2

The story of Pinocchio tells of a puppet whose nose grows whenever he tells a lie. We assume it does not grow when he tells the truth. Now here is a paradox involving the puppet Pinocchio, as it appeared in my email. For copyright reasons, I will have to describe the graphic.

A colourful Pinocchio is shown seated, and saying, "My nose will grow now." Underneath is written:

Epic paradox:

   Pinocchio says, "My nose will grow now."
   If he says it will grow, but it doesn't, he's lying.
   But it grows when he lies, so he would be
   telling the truth. But his nose still grew while
   he told the truth.

If you do not understand the given explanation (and I would not be surprised!), see below.

My version! …

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