The following Bongard problem was devised to be added to my book “Bongard puzzles for Kids”.
Which differences do you see between the 6 images on the left and the 6 images on the right? Yes, there is more than one difference. How many of them can you spot?
And more interesting perhaps: how many differences do you think there are? Or, more in terms of Bongard problems: How many rules are there which describe the 6 figures on the left, while non of the 6 images on the right comply with this rule?
Now think a while before you scroll down.
Oh, one last bit. A couples of differences are listed here.
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Actually, the number of rules is infinite. Surprise?
Let me try to show this. There are at least three ways to show that. All three ways need some math, but we’ll start with the easiest way. In subsequent posts I hope to move towards a more rigorous proof.
But we’ll take a bit of a detour.
Let’s start with a “list of numbers” puzzle, where you are asked to expand a short list of numbers with the next number. You often encounter them in many IQ tests.
Let’s have a look at this one:
1, 3, 9, 27, ….
What is the rule? What is the next number?
You probably answered ‘81’, didn’t you? You noticed that the rule seemed to be that every number was 3 times the previous number.
But is that the only rule that explains this sequence of 4 numbers? No, there are at least 2 other rules. One is, that every number is odd. So the next number might be 81, or 29, or even a lower number such as 7. A second rule that describes this sequence is that every number must be higher than the previous one. 81 would still be good, but so would be 28 or 29. And a third possible rule: the numbers 1, 3, 9, 27 are repeated over and over again.
These examples show that such a simple sequence of numbers has multiple solutions. But multiple solutions is not the same as infinitely many solutions.
How do we find infinitely many rules that explain 1, 3, 9, 27, …? Well, one possibility a variation of the third rule suggested above. Suppose the rule is that the numbers 1, 3, 9, 27, 31 are repeated over and over again. Or 1, 3, 9, 27, 28. Any number can be used as the fifth number. Arbitrarily? Absolutely! Ugly? Yes! Artificial? Yes! But nevertheless the four number sequence comply to these rules. You may object that these rules are ugly. You may object that these rules have no way to be predicted. And you are absolutely right. Only if we had a sequence 1, 3, 9, 27, 28, 1, 3 we would have a reasonable base tot suspect that there is a repeating group in this rule. With just four numbers, in ascending order, we have no base to suspect there is a repeating group. But neither can we exclude it. But for the four numbers we started with: they do comply with the rule.
So far for expanding a list of numbers. But there are still two missing steps. The first missing step is: What is the relation between the Bongard puzzle and a list of numbers? The second step is: What about the six diagrams on the right? Don’t the cross out those infinitely many rules?
As for the question how we get from Bongard problems to numbers: Let’s say each square is 100 x 100 pixels. A pixel can have 16 million colors. So the first pixel can have a 16 million values. And with 100 x 100 pixels, we have 10.000 pixels.
But we need something more.
We are going to map all 10000 pixels to a unique number.
Number the pixels p1 to p10000.
Now take
value of (p1) * 16M^0 +
value of (p2) * 16M^1 +
value of (p3) * 16M^2 +
…
value of (p10000) * 16M^9999
Yeah, they are pretty big numbers. Your life here on earth would be too short to count them, but there is nothing here our math can’t handle.
The second step which we so far did not touch, is the question how many rules are cancelled out by the 6 examples on the right. That is something for another post.
justpuzzles 