Can you solve it? The ants on a stick puzzle

Six ants are on a stick. Will you twig what happens next?

Hello guzzlers.

Many classic puzzles involve animals, such as the ones about a bird flying between trains, a fox and a goose crossing a river and four dogs chasing each other.

A recent addition to the genre involves ants on a stick.

Todays variation is by maths author Rob Eastaway, about whom more later. Also below is the final word on the Monday puzzle from two weeks ago in which I asked you to complete the equation 10 9 8 7 6 5 4 3 2 1 = 2016.

Okay, to todays puzzle:


A deck of ants: A, B, D and E are moving left to right. C and F from right to left. Illustration: Rob Eastaway

Four red ants and two black ants are walking along the edge of a one metre stick. The four red ants, called Alf, Bert, Derek and Ethel, are all walking from left to right as we look at the diagram, and the two black ants, Charlie and Freda, are walking from right to left.

The ants always walk at exactly one centimetre per second. Whenever they bump into another ant, they immediately turn around and walk in the other direction. And whenever they get to the end of a stick, they fall off.

Alf starts at the left hand end of the stick, while Bert starts 20.2 cm from the left, Derek is at 38.7cm, Ethel is at 64.9cm and Freda is at 81.8cm.

Charlies position is not known – all we know is that he starts somewhere between Bert and Derek.

So here is the puzzle: Which ant is the last to fall off the stick? And how long will it be before he or she does fall off?

Ill be back later today with the solution. (Solution now up)

Thanks Rob Eastaway for the puzzle. Robs new book Maths on the Go: 101 Ways to Play with Maths was out last week.


Two weeks ago the Monday Puzzle was to complete the equation 10 9 8 7 6 5 4 3 2 1 = 2016 using just the basic operations of addition, subtraction, multiplication and division.

Many of you submitted solutions, such as (10 x 9 x 8 x 7 x 6)/(5 + 4 + 3 + 2 + 1) = 2016. I also wrote that I was half hoping that a computer scientist would let me know the total number of solutions. Well, my wish was granted. Twice.

The eminent computer scientist Peter Norvig, Director of Research at Google, made a notebook on the problem and concluded that there were

44,499 different solutions.

A British programmer who uses the name Zefram concluded that there were:

8184 different solutions.

Norvig graciously said he agreed with Zeframs analysis over his own. One difficulty in comparing the solutions is because of different definitions of what counts as a distinct solution.

Still, there are a lot of solutions, something which may come as a surprise to the casual puzzle solver or non-mathematical reader of this column.

Zefram also worked out that 2016 was a relatively easy year to solve the problem. There are only 2868 expressions evaluating to 2015 and 3154 evaluating to 2017.

He also calculated that for the expression 10 9 8 7 6 5 4 3 2 1 using only +, , x and :

  • The largest uniquely-reachable positive number is 5443200.
  • The largest uniquely-reachable negative number is -937440.
  • The largest uniquely-reachable positive non-integer is 1209605/2.
  • The largest uniquely-reachable negative non-integer is -679680/7.
  • The smallest uniquely-reachable positive number is 1/40321.
  • The smallest uniquely-reachable negative number is -2/45423.
  • The smallest uniquely-reachable positive integer is 12262.
  • The smallest uniquely-reachable negative integer is -6907.

A question both men asked is what is the first year such that you cannot make with 10 9 8 7 6 5 4 3 2 1 and the four operations. They both agreed that the year was:


So the puzzle is good for a few millennia yet!

I post a puzzle here on a Monday every two weeks.

My most recent book is the mathematical adult colouring book Snowflake Seashell Star. (In the US its title is Patterns of the Universe.)

You can check me out on Twitter, Facebook, Google+ and my personal website.

And if know of any great puzzles that you would like me to set here, get in touch.

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