# Wild About Math!Making Math fun and accessible

20Jul/095

## MMM #37: More spiral fun

Our new Monday Math Madness extends the exploration of MMM #36.

Here's the problem:

Based on the introduction to spiral numbers presented in MMM #36, solve one (or both) of these problems:

1. Come up with an algorithm that tells what number is at an arbitrary X, Y coordinate.
2. Come up with an algorithm that tells the X, Y coordinates for an arbitrary positive integer.

I'll give three prizes for this contest, one to a random solver as usual, and one each for the best solution to each part of the problem. Even if you've won a prize in the last year you're eligible for one of the "non-random" prizes.

18Jul/091

## MMM #36: Spiral Numbers – Winner!

27 of you submitted solutions to MMM #36. Random.org has selected Mathias Malandain as the winner. Congratulations, Mathias!

This was the problem:

Imagine arranging the positive integers in a spiral pattern.
The numbers from 1 to 16 look like this in the spiral pattern.

```10  9  8  7
11  2  1  6
12  3  4  5
13 14 15 16
```

The location of each number corresponds to an X,Y Cartesian coordinate where the number 1 is at the origin: (0,0).
2 is at (-1,0). 3 is at (-1,-1). 4 is at (0,-1). 5 is at (1,-1). 6 is at (1,0). 7 is at (1,1) and so on.

What is the X,Y coordinate of the number 1,000,000?

Most people solved the problem by noticing where squares lie in the spiral.

5Jul/094

## MMM #36: Spiral numbers

I'll be contacting the three winners of MMM #35 in the next couple of days to get them their prizes.

Let's move on to MMM #36. I made this one up just for Monday Math Madness!

Imagine arranging the positive integers in a spiral pattern.
The numbers from 1 to 16 look like this in the spiral pattern.

```10  9  8  7
11  2  1  6
12  3  4  5
13 14 15 16
```

The location of each number corresponds to an X,Y Cartesian coordinate where the number 1 is at the origin: (0,0).
2 is at (-1,0). 3 is at (-1,-1). 4 is at (0,-1). 5 is at (1,-1). 6 is at (1,0). 7 is at (1,1) and so on.

What is the X,Y coordinate of the number 1,000,000?

4Jul/091

## MMM #35 Fibonacci fun – we have winners!

I'm giving away three prizes this time. Yes, I did say I would give four prizes but when I reviewed the first two submissions one of them was not correct. Henno Brandsma, editor for the Topology Q+A Board, sent the first correct solution, and that was the only correct solution submitted before I gave a small hint. I'm giving a prize to .mau. since he has sent in many solutions to MMMs and has never been selected by random.org. And, I'm giving a prize to Olivier, who was selected by random.org.

Here was the problem:

Let F(0)=1, F(1)=1, and F(n)=F(n-2)+F(n-1). This is the familiar Fibonacci series.
Simplify F(0)xF(1) + F(1)xF(2) + F(2)xF(3) + F(3)xF(4) + … + F(n-1)xF(n) + F(n)xF(n+1)

I got 14 submissions. Most of you sent induction proofs. Since induction doesn't give any insight as to how a solution was derived I can only guess that folks found patterns for sums as n increases then used induction to verify the patterns they saw.

Chao Xu submitted a nice induction proof at the Math Solutions Blog. The solution was password protected to not help others before the submission deadline. I've asked Chao to unprotect it.

S(n+1)=( f(n+2)^2 - f(n) x f(n+1) -1 ) /2

Note how this answer eliminates the need to consider odd and even cases. And, yes, the index is off by 1 in this formula but the insight is excellent.

Jacques Descartes had a clever simplification of the odd/even thing:

S(n)=F(n+1)^2-.5+.5(-1)^n

Do you see how odd/even is handled?

Matthias Malandain proved these two formulas:

S(2n) = (F(2n+1))^2
S(2n-1) = F(2n-1) x F(2n+1)

Another way to approach the problem was by looking at the pictures I posted. While my way does not provide a proof it gives one an intuitive sense of why the answer is what it is.

Watchmath submitted a nice proof that didn't use recursion.

I'm delighted to see the variety in solving this problem.

Stay tuned Monday for the next MMM, right here at Wild About Math!

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