In 1997, James Martin, a quietly unassuming Oxford maths PhD, entered television gameshow folklore with his stunning solution to a maths problem on Countdown.
The challenge: make 952 using the numbers 100, 75, 50, 25, 3 and 6. You may use the four standard operations of addition, subtraction, multiplication and division. You do not have to use all the numbers and you cannot use the same number more than once.
Oh, and you have 30 seconds.
James' solution has made it on to YouTube – well worth four minutes of your time:
The clip is television gold. As James’ solution dawns on co-host Carol Vorderman, she erupts into a fit of laughter that captures the sentiment of the studio audience and those of us watching on screen.
We’re all thinking the same thing. How the hell did he pull this off in 30 seconds?
Most people can get to 954 easily enough:
100 + 6 = 106
106 x 3 = 318
75/25 = 3
318 x 3 = 954
The brilliance of James’s solution is in how he negotiates the excess of 2:
- 100 + 6 = 106
- 106 x 3 = 318
- 318 x 75 = 23,850 (this is where Carol loses the plot – numbers this high are very rarely seen on the show)
- 23,850 - 50 = 23,800 (Carol is completely hysterical by now)
- 23,800 / 25 = 952
Looks pretty remarkable, right? And it is, but not for the reasons many people think.
If you watch the clip closely, you’ll notice that James does not actually know the value of 318 x 75. Here’s where the real magic lies – he doesn’t need to know.
James understands that, by combining steps 3-5, the overall effect is to multiply by 3 and subtract 2 (which is exactly what he’s after). You see, James is familiar with basic properties of numbers like distributivity (a fancy way of saying that 3 x (4+5) = 3x4 + 3x5, for example) and order of operations (BIDMAS, BODMAS, PEMDAS or whatever else it’s called these days).
James knows that, if he multiplies by 75 and divides by 25, it is the same as multiplying by 3 (heck, that’s what we did in the 954 solution). Similarly, he knows that 50/25 = 2. The key is to deftly combine these two calculations into three steps.
Good number sense
To do this in 30 seconds is still a hugely impressive feat. But it does not rely on supernatural mental-arithmetic skills. James surely had a command of mental maths, but his real talent lies in his number sense. He gets how numbers and operations interact with each other. He understands the structures that underpin numbers.
And he combines procedural skills with conceptual understanding to produce the perfect game-show moment.
Here’s the kicker: it does not require an Oxford maths PhD to develop number sense.
When I appeared on Countdown in 2008, a fellow contestant (my arch-rival, it would turn out) developed an excellent website for practice. In its early days, I would see many online players struggle with the numbers game. They could barely put together a sequence of operations, and certainly not in 30 seconds.
And yet, over time, something quite fascinating happened. These same players became expert in the kinds of solutions produced by James (many of them wreaked havoc on screen, dominating their own Countdown series). These players were not regurgitating fixed solutions – there are far too many permutations to make that viable. But they had a familiarity and comfort with arithmetic that allowed them to manipulate numbers flexibly and at speed. Practice was making all the difference.
Jo Boaler has led the charge among maths educators in calling for a broader emphasis on skills and understanding in the maths curriculum. Countdown is the strongest evidence I have in favour of making number sense a cornerstone of primary maths education. We could do worse than making games like Countdown a staple part of the curriculum, replacing the drudgery of fact-acquisition with fun, open-ended problems that stretch students’ minds.
We no longer need human calculators. Yes, multiplication tables are important – students will be hampered if they don’t know the basics. But any policy that forces students to learn these properties by rote is hopelessly out of touch with society’s needs.
We must nurture flexible minds that can apply number skills to solve novel problems.