My Favorite Math Problem

It is the right general idea, but don't try to express it in numbers. Just think that for each non-center point P there is a unique corresponding point P' such that the center point is exactly midway between P and P'. That gets you your pairs of places to work with without having to bring numbers into it.

For a coin placed at the center it covers any given non-center point Q if and only if at covers Q's corresponding point Q'. After the first player plays the center we then have that for every point that is not yet covered that point's corresponding point is not yet covered.

Then you have to show that once the center point has been covered by a coin any subsequent coin played that covers a point R cannot also cover R's corresponding point R'.

Then you can proceed with your pairing argument that whenever player 2 covers a set of points player 1 can cover the corresponding points on the next turn, and that after this we have restored the condition that for every point still available its corresponding point is also still available.

But yes, if it is clearly understood that the host always opens another door after you picked one, you should switch.

> 100 ants are spaced evenly on a 1m stick

Are the ends at the end of the line of evenly spaced ants on the very ends of the stick?

> They travel at 1m/minute.

Is each ant initially traveling? What is the initial travel direction of each traveling ant?

This matters because if the ants are all traveling in initially traveling then (assuming maximal even initial spacing) it would take 1 minute for the last ant to fall off, but if each ant were initially traveling towards the nearest end it would take about 1/2 minute.

Googling, I found almost the same problem except that the question asked was what is the minimum time that guarantees all ants will have fallen off no matter what their initial directions are.

First, a small defnition: let dsum(X) denote "sum of the digits of the decimal representation of whole number x".

In the modulo 9 congruence, 0 and 9 are equivalent symbols. 9 ≡ 0 (mod 9). Though distinct integers outside of the congruence they are indistinguishable elements of the congruence.

Observe that since 9 is congruent to 0, the number 10 is congruent to 1: 10 ≡ 1 (mod 9). Therefore, all powers of 10 are also congruent to 1: 1 ≡ 10 ≡ 100 ≡ 1000 ... (mod 9).

Next, note that these numbers x ∈ {1, 10, 100, 1000, ...} all have the property that x ≡ dsum(X) (mod 9). The sum of decimal digits of each power of 10 is 1, and each power of 10 is congruent to 1, modulo 9.

Within a congruence, whenever we multiply by any congruence element that is indistinguishable from 1, we get the same element. So all of these numbers 1, 10, 100, ... are identity elements in the modulo 9 congruence.

Identity means that, for instance 7 ≡ 70 ≡ 700 ≡ 7000 = 7000 ... (mod 9). If we multiply the congruence element 7 by any power of 10, the result is congruent to 7, modulo 9, because 10 is the identity element: no matter how many times we multiply an integer by 10, the resulting integer is the same modulo 9 congruence element. (Moreover, also note that dsum(7) = dsum(70) = dsum(700) ...).

Next, note that every decimal integer is formed by a sum combination of multiples of power of 10.

For instance, 1234 is equal to 1 x 1000 + 2 x 100 + 3 x 10 + 4.

But suppose we do this arithmetic in the modulo 9 congruence: it must necessarily be that:

1000 + 200 + 30 + 4 ≡ 1 + 2 + 3 + 4 (mod 9).

This is because, individually, 1000 ≡ 1 (mod 9), and 200 ≡ 2 (mod 9), 30 = 3 (mod 9). The power of 10 factors do not matter, since they are really powers of the identity element 1 in the mod 9 congruence.

The succinct explanation is then: every whole number has a decimal form made up of digits, which are multiplied by powers of 10. But in the modulo 9 congruence, powers of 10 do not matter: they map to the multiplicative identity element 1. That leaves the sum of the digits indistinguishable (in the modulo 9 congruence) from the value of the number (which is also a kind of sum of the digits, with each digit being scaled by a different power of 10).