5 ms·

My process. 1) Ugh, they cheated. How dumb. 2) No, wait, that's clever. I like it. 3) Oooh, it's that perennial solution of thinking with orthogonality. 4)

My process. 1) Ugh, they cheated. How dumb. 2) No, wait, that's clever. I like it. 3) Oooh, it's that perennial solution of thinking with orthogonality. 4) Wait, what? The golden ratio's in there?! No. Way! That's so cool. And the solution it's some random garbage, but pretty regimented. Neat-o!

randtrain34 5y ago
This has to be one of the cooler instances of the golden ratio.
- IngoBlechschmid 5y ago
  Another is Conway's Soliders: A variant of peg solitaire, it takes place on an infinite checkerboard. The board is divided by a horizontal line that extends indefinitely. Above the line are empty cells and below the line are an arbitrary number of game pieces, or "soldiers". As in peg solitaire, a move consists of one soldier jumping over an adjacent soldier into an empty cell, vertically or horizontally (but not diagonally), and removing the soldier which was jumped over. The goal of the puzzle is to place a soldier as far above the horizontal line as possible. (https://en.wikipedia.org/wiki/Conway%27s_Soldiers https://en.wikipedia.org/wiki/Conway%27s_Soldiers) How many soldiers have to be put below the line before the game starts to enable at least one soldier to reach a given height over the line? For height 1, you need 2 soldiers. For height 2, you need 4. For height 3, you need 8. For height 4, you need ... 20. For height 5, it's impossible, by the golden ratio.
  - jerf 5y ago
    In one of Numberphile's more in-depth videos (41 minutes), they have a full video proof of this, including the derivation of where the golden ratio comes from: https://www.youtube.com/watch?v=Or0uWM9bT5w&t=10s https://www.youtube.com/watch?v=Or0uWM9bT5w&t=10s
  - tomerv 5y ago
    > For height 5, it's impossible, by the golden ratio. Strictly speaking, it's Conway's proof uses the golden ratio. But it could be that there's an alternate proof that doesn't use the golden ratio.
    - IngoBlechschmid 5y ago
      Yes, you are right! In fact, to people working in foundations of mathematics it is routine to compile away any explicit use of the golden ratio in Conway's proof and thus obtain a proof which only refers to the integers. However the resulting proof will be less perspicuous. I don't know of any transparent proof avoiding the golden ratio, though it definitely could exist.
rndphs 5y ago
> Wait, what? The golden ratio's in there?! No. Way! That's so cool. And the solution it's some random garbage, but pretty regimented. Neat-o! The golden ratio is the solution to x(x-1) = 1. And its reciprocal is the solution to x(x+1) = 1. This equation is probably the simplest equation with irrational solutions aside from x*x = 2 and hence appears all over the place in mathematics.
cbsmith 5y ago
I had a very similar experience.

randtrain34 5y ago

This has to be one of the cooler instances of the golden ratio.

IngoBlechschmid 5y ago
Another is Conway's Soliders: A variant of peg solitaire, it takes place on an infinite checkerboard. The board is divided by a horizontal line that extends indefinitely. Above the line are empty cells and below the line are an arbitrary number of game pieces, or "soldiers". As in peg solitaire, a move consists of one soldier jumping over an adjacent soldier into an empty cell, vertically or horizontally (but not diagonally), and removing the soldier which was jumped over. The goal of the puzzle is to place a soldier as far above the horizontal line as possible. (https://en.wikipedia.org/wiki/Conway%27s_Soldiers https://en.wikipedia.org/wiki/Conway%27s_Soldiers) How many soldiers have to be put below the line before the game starts to enable at least one soldier to reach a given height over the line? For height 1, you need 2 soldiers. For height 2, you need 4. For height 3, you need 8. For height 4, you need ... 20. For height 5, it's impossible, by the golden ratio.
- jerf 5y ago
  In one of Numberphile's more in-depth videos (41 minutes), they have a full video proof of this, including the derivation of where the golden ratio comes from: https://www.youtube.com/watch?v=Or0uWM9bT5w&t=10s https://www.youtube.com/watch?v=Or0uWM9bT5w&t=10s
- tomerv 5y ago
  > For height 5, it's impossible, by the golden ratio. Strictly speaking, it's Conway's proof uses the golden ratio. But it could be that there's an alternate proof that doesn't use the golden ratio.
  - IngoBlechschmid 5y ago
    Yes, you are right! In fact, to people working in foundations of mathematics it is routine to compile away any explicit use of the golden ratio in Conway's proof and thus obtain a proof which only refers to the integers. However the resulting proof will be less perspicuous. I don't know of any transparent proof avoiding the golden ratio, though it definitely could exist.

rndphs 5y ago

> Wait, what? The golden ratio's in there?! No. Way! That's so cool. And the solution it's some random garbage, but pretty regimented. Neat-o! The golden ratio is the solution to x(x-1) = 1. And its reciprocal is the solution to x(x+1) = 1. This equation is probably the simplest equation with irrational solutions aside from x*x = 2 and hence appears all over the place in mathematics.

cbsmith 5y ago

I had a very similar experience.