Pre-school mathematics

29.03.2012 20:02

Recently the following question has been circling the Internet and also ended up in my inbox a few times. It's yet another one of those silly exercises that ask you to continue a sequence based on a number of examples. They are a pet peeve of mine and I'll explain why I hate them here once and for all instead of replying to everyone that sent a copy my way.

This problem is said to be solved by pre-school children in minutes

The logic of this specimen is particularly broken, even putting aside the questionable statement about pre-school children being able to solve it in mere minutes. It's obvious that whoever wrote that is not familiar with the meaning of the equals sign, as 8809 isn't equal to 6. So strictly reading, this exercise gives you 21 false statements. From that it can be simply inferred that the author is looking for another false statement, so ??? can be replaced with any number not equal to 2581.

With that out of the way, let's look at what was probably meant with this notation. You are given N=21 pairs of values (xi, yi) such that:

f(x_i) = y_i \qquad i\in[0 \dots N-1]

And asked to find yN for a given xN:

f(x_N) = y_N

From a purely mathematical standpoint, this problem has an infinite number of solutions. To demonstrate, assume that f is a function that satisfies the condition set above. In that case:

f(x) + g(x)


g(x_i) = 0 \qquad i \in [0 \dots N-1]

also satisfies this condition. Note that g is a function that can have an arbitrary value at xN and hence this new solution will give a different value of yN.

Just to make this a little bit more practical, here's an example of a polynomial that goes through N points:

f(x) = \sum_{i=0}^{N-1} (x - a_i) \prod_{j=0; j \neq i}^{N-1} (x - x_j)
a_i = x_i - \frac{y_i}{\prod_{j=0;j \neq i}^{N-1}(x_i - x_j)}

And here is how its graph looks like if you plug in the given 21 values:

Polynomial function that goes through given 21 points

At x=2581 it gives the value of around 5.8·1073.

Note that using the formula above and N=22, you can get a polynomial that goes through the required 21 points plus the 22nd that you can choose to your liking.

This is just one example. You could construct functions that meet the requirements from trigonometric or exponential functions in a similar way.

Sure, so this is probably not the solution the author of this question was looking for. You could say that I'm overcomplicating things and that the rule that pre-school children would come up with, based on the shape of the digits the author arbitrary chose to represent these numbers in, is nicer. But there is no such thing as niceness in mathematics (see interesting number paradox).

This is the big difference between mathematics and physics. Our universe has physical laws that for some reason can often be represented in certain mathematical forms. If a liter of water has a mass of one kilogram and two liters a mass of two kilograms it's quite likely that three liters will have a mass of three kilograms and so on. But this is purely based on our experience and beliefs about our environment and not on any purely mathematical basis. Such a prediction, just like the one above and many like it, when stripped of any physical context, does not make any sense.

Posted by Tomaž | Categories: Ideas


I liked this challenge as I solved it quickly at 68! Can I see IJS influence on you YM? When I gave a compliment to one young EE PhD there, good technicians commented: He doesn't know how transistor works! I am glad you do. Regards from warm Jagodina, Serbia. I had a flu for a week but solved my spectral problems with orthogonal FFT detection of narrow Morse in impulse noise.

Posted by MMM

The answer is 2.

You are over-complicating things but not because you're doing anything wrong. It's set up to look like a math problem at first glance... but it's not. It's a logic puzzle and the reason pre-schoolers can get it is because they don't know all those math rules, some don't know any math at all, and aren't looking at it as a math problem.

Posted by anon

well, i know that the answer is 2, but i spent so long trying to figure this out and i had no idea what it was! So i have no idea how the answer is 2? how? why? i am so confused!

Posted by hurmp

Number of circles in each 4 digit 'number'.

Preschoolers have no sense of numbers, so they'd only see them as shapes.
So the number 0 has 1 circle.
1 has none
2 has none
3 has none
4 has none
5 has none
6 has 1 circle
7 none
8 has 2 circles
9 has 1 circle

Hence the 'number' 2581 is just a collection of shapes with 2 CIRCLES in it.

Therefore, answer is 2.

Posted by Nikhil

While I agree with the idea that the = can interpreted as a mathematical function, that is part of riddle. It is an example of functional fixedness. Once we learn one way to think of a problem or an object we have great difficulty changing the way we think. An = is an arbitrary symbol that you are taught to mean "equals." The symbols that we know as numbers are also arbitrary, you see the symbol "8" and you immediately interpret at "the number eight" rather than two circles. Pre-school aged children are less likely to have been taught that "8" is the universally agreed upon symbol for "the number eight."

Posted by Lauren

The answer is 2.

the number of circles has nothing to do with it. If 0000 = 4, then it stands to reason that 0 = 1. If 1111 = 0 then again it stands to reason that 1 = 0. If you follow the sequence for the give values, you will note that 0 = 1, 1 = 0, 2 = 0. 3 = 0, 5 = 0, 6 = 1, 7 = 0, 9 = 1. You will note that 4 does not exist, that leaves just 8 to find. First of all test it out on a few combinations, say 9313 and 5531. Assign the above numbers to their respective values, and you get:

9313 = 1+0+0+0 = 1
5531 = 0+0+0+0 = 0

Use a sequence with 8, and do some simple maths:

8193 = ?+0+1+0 = 3 or 3 - 1 = 2, therefore 8 must be equal to 2.

Test it with the following given sequences with 8:

8809 = 2+2+1+1 = 6
8096 = 2+1+1+1 = 5
9881 = 1+2+2+0 = 5

Therefore the problem is solved thus:
2581 = 0+0+2+0 = 2.

Posted by j

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