Is it possible to combine creative and critical thinking? Yes, actually.

Critical thinking and creative thinking are often taken to be opposites. You can be good at one, but not the other. Or, a task that requires you to use one, won’t require you to use the other.

I suspect this is less the case than we think. For example, there is at least one intellectual discipline that combines both creative and critical thinking almost all the time. That discipline is mathematics.

Have a look at the following short video, from the new film “X+Y“. It shows the construction of a typical elegant mathematical proof. The problem is posed. There is a feeling of blankness and frustration. And then there is a flash of insight – the key creative thought. That insight is then followed by logical deductive reasoning – the work of critical thought.

Asa Butterfield‘s character, Nathan, is initially at a loss. But then the moment of creative insight hits him: don’t regard the problem as cards, regard the problem as binary numbers. From there, once the link to binary numbers is made – and there is no way to make that link deductively, it is simply a flash of insight – the way is open for the rules of logic to take their course. It is now easily demonstrated that the binary numbers generated by the given rules of card turning will always be decreasing. At some point the series will hit zero and therefore terminate.

This rhythm – a creative insight to move the problem into a new domain – then followed by the application of the rigorous rules of logic – is typical of mathematics. Again and again you see mathematicians both use creative and critical thought in the one proof. As such I think it’s fair to say that in at least one human discipline creative and critical thought work together.

If you’re interested, you can read on for the detail of Nathan’s proof. In any event, I would love it if you could subscribe to this blog, to receive updates for each new post, by clicking on the “subscribe” button in the top right hand corner.

The details of Nathan’s proof

The problem is this. You have a sequence of cards, all face down. A permitted move on this sequence is as follows. You can turn one face down card, face up. But you must also turn over the card immediately to the right of it (if that card was face up, it’s now face down, and vice versa). Show, that if you keep applying this move to the sequence that eventually you will have to stop.

Nathan’s creative insight is elegant. Regard each face down card as a 1 and each face up card as a 0:

• Initially you will have a sequence of 20 “1s”: 1111…111.
• Over time, as you apply the rule, you might end up with a sequence like this: 1…10011010. You’ve turned cards face up (turning them into 0s, and other cards face down, turning them back to 1s).
• But all these unfolding 20 digit sequences of 1s and 0s are simply binary numbers (long ones, but binary numbers just the same).
• And the insight is that the rules will always operate so that the 20 digit binary numbers at each step are decreasing. That is, when you apply the rule to your 20 digit binary number, the next 20 digit binary number you get will always be less than the one you just operated on.

Why is that so? Let’s do a quick revision on binary numbers.

Remember, a binary number has a “place value” like a decimal number, but instead of that place value being a power of 10, it’s a power of 2.

So, in binary:

• 0001 = 1 (ie 0 x 2^3 + 0x2^2 + 0x2^1 +1×2^0 = 1) (remember that a number raised to the power of zero is 1).
• 0010 = 2
• 0011 = 3
• 0100 = 4
• 0101 = 5
• 0110 = 6
• 0111 = 7
• 1000 = 8

Now let’s get back to Nathan’s proof:

• Nathan suggests a sequence with this component in it: 10011010.
• You could use your rule to operate on the middle two 1s: 10011010. The rule will turn the first bolded 1 into a 0 (the rule says, choose a face down card, a 1, and turn it over, a 0). It will turn the second bolded 1 into a 0 as well (turn the next card to the right over). We now have 10000010, which is clearly a lower number than the first number, because of the way place value is working.
• You could then use the rule to operate on the last 10 in the new sequence. The 1 (choose a face down card) becomes a 0 and the next 0 to the right becomes a 1. Giving 10000001. This number is lower again.
• It should be clear that this process will continue to deliver numbers that are strictly decreasing.
• Eventually, you will hit zero (all 0s) and you will have no face down cards to turn over. At that point you terminate.

So that’s it. Both a flash of insight and then a rigorous argument. Two different types of thinking in the one proof!

And, if you have come this far then you are just the type of person to appreciate the old maths joke: There are 10 types of people in this world – those who understand binary and those you don’t!