I came up with a multi-part math problem for the late primary or early high school level. I like this problem because it only requires knowledge of addition and multiplication, yet it's not that easy at all.
Introduction: in this task, we will be drawing various line segments. Usually, in geometry, we label segments with letters like "a", "b", "c"... In this task, the labels will be numbers: "1", "2", "3"... Meaning, instead of marking segments with letters, we will number them.
Part zero (introductory).
Instruction: draw any three line segments, labeled "1", "2", "5".
Comment: the point of this part is to make sure the victim understands that a segment's label is not the same as its length (that these don't have to be segments of lengths 1, 2, 5).
Part one:
draw segments labeled "1", "2", "4", such that:
- combining segments "2" and "2" gives segment "4" (meaning that (length of segment "2") + (length of segment "2") = (length of segment "4"))
- combining segments "1" and "2" gives segment "2"
- combining segments "1" and "1" gives segment "1"
That is, so that adding lengths works like multiplying labels.
Comment: Someone who is more mathematically advanced tends to overcomplicate things with logarithms at this stage. It's possible, but it can also be done without thinking about logarithms, just by playing around for a moment. And, for example, getting this solution:
- segment "1" has length 0
- segment "2" has length 2
- segment "4" has length 4
Yes, this is simply the function "f(x) = 2 * log(2)(x)", but you don't need to think about logarithms here at all.
Part two:
draw segments labeled "1", "2", "3", "4", "6", such that:
- combining segments "1" and "1" gives segment "1"
- combining segments "1" and "2" gives segment "2"
- combining segments "1" and "3" gives segment "3"
- combining segments "2" and "2" gives segment "4"
- combining segments "2" and "3" gives segment "6"
So once again: adding lengths should act as multiplying labels, we just have more labels and more segments.
Comment: a mathematically advanced person who approaches this problem thinking about logarithms will get irrational results here - and might not even notice that a solution with natural lengths exists. Someone at the late primary or early high school level might, for example, find this solution:
- segment "1" has length 0
- segment "2" has length 2
- segment "3" has length 3
- segment "4" has length 4
- segment "6" has length 5
Part three:
and now the other way around: I will give you the lengths of the segments, and you have to choose the labels so that adding lengths results in multiplying labels. And the segment lengths are: 0, 1, 2, 3, 4, 5.
Comment: well, this time an advanced person who knows logarithms will have an easy time (1, 10, 100, 1000, 10000, 100000) - while a regular person will have to figure things out a lot more before discovering this.
I like this problem because it provides good mental gymnastics: you have to avoid getting lost between what is a label and what is a length (both are numbers), what we add, and what we multiply.