# Number play

One of our goals in creating BrainQuake was to leverage kids’ (and adults’, ourselves included) use of video games to arouse interest in, and enjoyment of, mathematics. Not because math is useful (though it is), but simply because it can be fun.

We know from feedback we have received from people who have used our product that it has succeeded in doing that. The video below shows a young girl completing — with her father’s help — one of the more difficult Gears puzzles (in its original presentation as the Wuzzit Trouble mechanic, now one of three puzzles that make up the BrainQuake app).

When that child completed the solution, she had in fact solved a system of simultaneous linear equations in three unknowns. [It’s worth noting that, although she needed her father’s help for this one, she did fine on simpler ones with only one or two unknowns.] But that was not her motivation. Indeed, she had never seen systems of simultaneous equations, except when presented as a fun puzzle like Gears. Nor could she have automatically solved such a system if presented with it using the familiar symbolic language of algebra. But when she eventually does get to algebra (by now she surely has), she would already have the experience of playing with linear equations in a video game. Skills acquired in games can turn out to be useful in different contexts later on in life. (Obvious examples are pilots learning to fly in flight-simulators and soldiers learning to fight in battlefield simulators. Interestingly, versions of both those technologies are marketed as video games.)

Using video games and puzzles in education in this way (game-based learning) might seem like a new idea, but the only thing that is new is the video-game part. Using games and puzzles in general has a long history in education, particularly to arouse early interest in mathematics. Here is an example of a numerical curiosity that aroused my interest when I was at school. It doesn’t require any fancy technology, though a cheap calculator can come in handy.

It concerns the number 7, which turns out to have a rather peculiar property. If you work out its reciprocal as a decimal, the result is a pattern of six digits which repeats itself ad infinitum:

1/7 = 0.142857 142857 142857 …

This is where it gets interesting. If you multiply this number by any of 2, 3, 4, 5, 6 (to obtain the decimal representation of 2/7, 3/7, 4/7, 5/7, 6/7, respectively), you get the same pattern shifted along. For example:

2/7 = 0.2847 142857 142857 …

3/7 = 0.42847 142857 142857 …

Cute, no? But is there anything to it?

What happens when you form the reciprocals of 2, 3, 4, 5, 6, 8, and 9? What kind of patterns do you get?

What if you go beyond 9? Are there any other numbers N beyond 10 such that the decimal representation of 1/N consists of an infinite repeating pattern which is reproduced (up to a shift) when the number is multiplied by each of the numbers 2 to N–1?

Can you find any such numbers N less than, say 50, that behave like 7, other than 7 itself?

It’s possible to attack this problem mathematically (give it a try), but the simplest way to start is to make use your favorite calculating device. Work out reciprocals and look for repeating patterns.

Each time you find a number whose reciprocal is a repeating pattern, compare it with the patterns you get by multiplying by 2, 3, etc. (An alternative way to perform this check is to repeatedly add your reciprocal to itself the required number of times, checking for a repeat pattern each time.)

Can you spot some sort of general rule which governs the behavior of numbers with this curious property?

I still have fond memories of going through this exploration by hand-calculation when I was a school student back in the early 1960s, before electronic calculators were available. (I did not go up to 50, I should add! I’d guess I probably stopped around 20.) What made it enjoyable was figuring it all out for myself.

If you have spent any time with the BrainQuake app, you will surely have discovered that it’s not a matter of just pushing buttons. It’s not a calculating device. It’s a platform on which you can explore mathematical puzzles, just as I did with the 1/7 curiosity.

My point is, (digital-) game-based learning is just a new variation of a way to arouse interest in — and achieve learning of — mathematics that goes back throughout the history of mathematics.

In that respect, it’s nothing new.

On the other hand, the use of digital games brings a number of benefits that cannot be obtained any other way. Effectively breaking the symbol barrier, as BrainQuake does, is one of them I have discussed in this blog. So too is real-time formative assessment at scale (at least for learning games that can do that, as ours can). And there are others that I will be talking about in the weeks to come.

Meanwhile, have fun with numbers!

– Keith

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