Why is this 15-Puzzle Impossible? - Numberphile

Don't try this at home - it's impossible... Professor Steven Bradlow explains.
More links & stuff in full description below ↓↓↓

NUMBERPHILE

Video by Pete McPartlan and Brady Haran

[rc]

The animation of the puzzle is very well done, as usual the animations are always great and help us to follow along with what the mathematician is saying.

[FrankSerio]

As a teen, I had a larger (7x7) version of the puzzle. It featured a picture of the Leonardo da Vinci's Vitruvian Man as the image. It turned out that there were two completely blank tiles which just happened to have opposite parity. So if you put the wrong identical tile into position, it was impossible to finish. It took me days to work that out.

[jpe]

When I was a kid I was one who was taught that these puzzles are called “Sam Loyd puzzles” because he invented them, but (quite recently!) I learned the truth that the puzzles predate the man, so I thank you, Numberphile, for reinforcing (confirming?) that information for me.
As an aside, when I was a little boy my cousin, who was in her early 50’s at the time, would impress me by solving 15 puzzles in the fewest moves and fastest times, and even now when she is a few days shy of her 96th birthday and suffering with dementia she can still solve these puzzles.

[disgruntledtoons]

Fun anecdote: Sam Lloyd was not able to patent the puzzle because he could not submit a solution to the challenge.

[PerfectlyNormalBeast]

Logic is so cool. Take a complicated looking problem, break it down, then prove all sorts of things in that simplified mental space

[zozzy]

"1, 2, 3, 5, ..." I think he's been working too much with the Fibonacci sequence lol

[asod]

Alternate title: Math Professor Forgets The Number 4
4:06

[jaapsch]

Instead of explicitly counting the lefts/rights/ups/downs for the blank space, I prefer giving the tray a checkerboard colouring. The blank space then changes colour every move, so must do an even number of moves to get back to the same colour (or same location).

[SirAndras]

This professor is amazingly easy to understand and listen to. Seems like a calm dude as well and can take his time to explain something. Nice vid!

[ingwermoschus]

His way of writing "odd" is rather 1(mod2).

[johanwilhelmsson]

We had one when I was young where the reverse order was labeled "unmöglich". (Presumably the puzzle was from Germany. :) ) I always wondered why, back then.

[amandawitt]

I graduated with a math degree about five years ago and don’t get to do math everyday like I did in school. The COVID-19 pandemic and shelter in place has given me an opportunity to pull out my old textbooks and think about things I love so much again.

I had my abstract algebra book out last weekend and was reading and thinking about permutations and parity.

Thanks for sharing this. Made my heart pitter patter. ❤️

[iolol]

wait this is my professor?? I saw the cover and was like "oh he looks familiar..."

[fluffycritter]

It took this video to make me finally understand where the word "parity" comes from - it's whether things are all "paired" up!

[Angi_Mathochist]

I actually discovered this and wrote a proof myself in high school. The assignment was actually to write a computer program to find the best next step in the puzzle. This was the early 1980s and we were learning Apple Basic. I fiddled with the example arrangement given and realized it couldn't be solved. So my immediate question was: which arrangements can be solved ash's which can't? I wrote a proof of why the given arrangement could not be solved and of which arrangements in general could and could not. (I had to invent my own language, bear in mind, because I was in high school and hadn't been exposed to most of the math involved yet.) And for my computer program, I wrote an algorithm to test whether the given arrangement was solvable at all, since setting your computer on an unsolvable problem would only serve to waste a lot of computing time until you gave up and aborted the program. :)

[Tobi]

That was amazing. I had such a puzzle as a kid and soon found out, that you can't have every configuration, but didn't know why. Now I know, i'm so happy :)

[JanischMaximilian]

The back of the puzzle was clearly designed by Matt Parker

[howardli]

Just read a book called "Tales of Impossibility", the puzzle from Sam Loyd was also discussed on Chapter 2. I am amazed of this kind of mathematical proof for its impossibility. Thanks Numberphile!

[Bibibosh]

23 minutes of utter joy and happiness!

Thank mathematics.
Thanks YouTube.
Thanks world!

[brianwestley]

There's a more devious version -- instead of 1-15, the tiles say "RATE YOUR MIND PAL" in horizontal rows. What you can do is show someone what it says and scramble the letters while they watch and challenge them to unscramble them. But what you do is put the "R" in YOUR into the upper left corner where the "R" in RATE should be. But you can't unscramble it that way -- the best you can do is RATE YOUR MIND PLA until you swap the Rs back.