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Brain-wrinkling puzzles from Ian Stewart

Matthew Killeya, contributor




Test your brain with a selection of the book's puzzles below.

Book Information:
Professor Stewart's Hoard of Mathematical Treasures by Ian Stewart
Profile Books Limited, £12.99
To be published in the US by Basic Books in April

1. Target Practice



Friar Tuck and Robin both fired a number of arrows.




Naturally, Robin pointed out the fallacy... but:

Which rings did the two archers hit?




Check your answers below

2. Accounting the Digits

In the Great Celestial Number Factory, where all numbers are made, the accountants keep tabs on how many times each digit 0-9 is used, to make sure that there are adequate stocks in the warehouse. They record these counts on a standard form, like this:


So, for example, the digit 4 occurs 3 times so Nugent writes '3' in the lower row of boxes, underneath the printed 4. Numbers are written so that they end in the right-hand box, like the example, and leading zeros may or may not occur. (None of that matters for this puzzle, but people do worry...)

One day Nugent was filling in the form, as usual, when he suddenly noticed something remarkable: the two numbers (that is, sequences of digits) recorded in the two rows of boxes were identical.

What was the number concerned?

Check your answer below

3. Digital Century

Innumeratus wrote the nine nonzero digits down in order, with gaps, like this:

1 2 3 4 5 6 7 8 9

'I want you to--' he began.

'--make 100 by inserting standard arithmetical symbols,' said Mathophila. 'That's easy, it was in Professor Stewart's Cabinet of Mathematical Curiosities, which you gave me for Christmas, but it goes back a lot further than that.' And she wrote:

123 - 45 - 67 + 89 = 100

'No, that's cheating,' said Innumeratus. 'I left gaps! You can't consider 1 2 3 to be one hundred and twenty-three, and--'

'Oh. No concatenation of symbols allowed, then.'

'Yeah. No caterwaulification--whatever.'

She thought for a moment, and wrote down

(1+2-3-4) × (5-6-7-8-9)

'Sorry, no brackets,' said Innumeratus.

Mathophila shrugged, and wrote

1 + 2×3 + 4×5 - 6 + 7 + 8×9

'You don't mind me using the rule that × precedes +, so I don't need to put brackets round individual multiplications, do you?'

'No, that's OK. But--uh--look, sorry, but no subtraction symbols either.'

There was a silence. 'I'm not sure that's possible,' said Mathophila.

'Wanna bet?' asked Innumeratus smugly.

What should Mathophila do?

Check your answer below


1. Tuck hit the outer ring (light grey), while Robin hit the three inner rings (dark grey).


The areas of successive circles are πr2 where r = 1, 2, 3, 4, 5 respectively:

π 4π 9π 16π 25π

The areas of the rings are the differences between these:

π 3π 5π 7π 9π

These are π multiplied by consecutive odd integers. Robin's integers are less than or equal to Tuck's, since Robin's arrows are closer to the centre. The two sets of odd integers must have the same sum. The only possibility is 1+3+5 = 9.

Bonus point: Six rings. The sixth ring has area 11π, so 1+5+7 = 11 is a second solution.

Further bonus point: Seven rings. Robin's odd numbers must be consecutive, and so must Tuck's. The next ring has area 13π, and 3+5+7+9 = 11+13 is a second solution with consecutive odd numbers.

2. This is the only number that works:


3. She should write down

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8×9

to avoid losing her money.

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For question 3 you could also have
1+2x3x4x5/6+7+8x9 = 100


Nice questions!


Does "i" count as a symbol?

1 + 2×3 + 4×5 + 6*i*i + 7 + 8×9

Or, without "cheating" provided one uses the math symbol for sqrt... (avoiding brackets thereby)

1+ sqrt(2^3^4) + 5 + 6 + 7 + 8 + 9


Q3. What about 1 x 2 x 3 x 4 + 5 + 6 + 7 x 8 + 9 = 100 ?


Question 1 is broken. Or at least badly worded.

A valid answer is that both archers got all their arrows in the centre circle. This could still satisfy the conditions "All of Robin's arrows are closer to the centre" and "The total areas hit by both archers are the same" (since they both hit only the same circle)


Geez, talk about mind wrinkling. I pondered the first one for about 10 minutes before giving up. I do agree, having looked at the answer that the first puzzle is badly (deceptively) worded.

What resources did these puzzles come from incidentally?


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