 # The Ice Cream Puzzle

## Some puzzles are harder than they look

Amy, Becky and Charlie love ice cream.  And being sociable girls, they like to eat ice cream together.

When Amy and Becky eat a tub together, they finish it in four minutes.  Amy and Charlie together take just three minutes.  And Becky and Charlie take a mere two minutes as they wolf it down.

How long does it take the three girls to finish a tub when they eat together?

Puzzles like this require a complete suspension of reality of course.  It's only possible to tackle the puzzle if you assume that the rate of eating ice cream is constant for each girl.  Often this same puzzle is posed in the form of a bath with three different plugs, but the human element makes it somehow more accessible.

[You might want to try the puzzle before reading on]

Presented with the ice cream puzzle, most teenagers are able to have a stab.  Those that are comfortable with algebra will probably even start setting out some equations.

Typically, you'll see this (from a teenager, a university student, or indeed a mathematically literate adult):

A + B = 4    A + C = 3   B + C = 2

"Aha" goes the thinking. "These are simultaneous equations, I can solve those".  Take the second equation from the first, we get B-C=1, add that to the last equation to get 2B=3, so B=1.5, making C 0.5 and A 2.5.  Which means that A+B+C=4.5.  Easy.

Except that there should be alarm bells ringing.  What this answer means is that when the three girls eat the ice cream together, it takes them 4.5 minutes, which is longer than it takes A and B (or any other combination) when they are a pair.  How can this be possible?

The reason why this answer is nonsense is that the equations above, that look so plausible, are themselves nonsense.  What does "A+B=4" mean?  The time A takes to eat a tub of ice cream plus the time B takes is equal to 4 minutes?  It can't mean that, it would mean that (say) A finishes a tub in 2 minutes and so does B in which case it can't take the two together longer than that.

It turns out this is a question about RATES, and these questions are fiendishly deceptive.  A similar example is working out average speeds.  If you travel to Brighton at 20 mph and return at 30mph what is your average speed?  25mph?  That would mean if you travel at 60mph and return at 0mph your average speed is 30mph...yet you never return, since you travel at 0mph!  (The correct average speed for 20out/30return is 24mph, as it happens.)

Here's one way to solve the ice cream puzzle.  Let's call the rate at which Amy eats ice cream as 'a' tubs per minute, and same for 'b' and 'c'.  We know that Amy and Becky take four minutes to eat one tub, so 4a + 4b = 1.  Likewise 3a+3c = 1 and 2b+2c=1.  Solve those three equations and you get a=1/24, b=5/24 and c=7/24, so between them the girls eat 13/24 of a tub per minute, and take 24/13 minutes (or 1 minute 51 secs) to eat the ice cream.

What's worrying is not that students struggle to get this answer (it's hard!), it's that so few who come up with an answer of over 4 minutes stop to think whether this could possibly be right.  The ability to ask "does this make sense?" is one of the most important skills we need in the next generation.