## 100 thoughts on “Pebbling a Chessboard – Numberphile”

1. Get Milked says:

I don't like this game.

2. Syafiq Roslan says:

Profesor Zvezdelina is so cute

3. Xaplomian says:

"The first number is a". and I have a favourite colour of the alphabet.

4. mama und papa says:

also, the maximum of possible clones in the bottom row and leftmost colum is two, so plenty of greenland unfilled

5. Francis Lai says:

7:30

6. Lanie W. says:

I'm in middle school and I would have said the sum of the first row is two

7. liam lynch says:

vsause did a video on supertasks which relates to this

8. Ryan says:

Don't free the clowns!!

9. Ale Rashid says:

Why did we asume the original mass is 2? (its like starting with 1 pebble and associating it with the mass number 2, even though we started with 3 pebbles) If there were 3 clones originally, may i start the mass at 3? Or maybe even 6?

10. Jehovaservant _ says:

It can't be done, even in an infinite # of moves, because the board still has a corner, and you can't put a clone in the corner.

11. NetAndyCz says:

Is it possible to free the clones in INFINITELY many moves?

12. Kasran Fox says:

Free the Clowns

13. Aiden Ocelot says:

"Let's use the very first number you learned in school"
Seven!
"One"
Darn it!

14. amante pensanta says:

Oh, those poor clown clones…

15. bugoobiga says:

1:17 the right what?? bart fans?

16. Xsedim says:

Chezzboard

17. Jackson DeStefano says:

I tried to do this using a spreadsheet. You think it's simple enough, you get the first two out pretty easily, but the last one is desceptively tricky. You see that in order to get space for the clone, you need to duplicate this clone, but that is blocked by a third clone. Okay, so you duplicate the third one, then the second one, but then you realize that now the top part is blocked and you need to move that, so you go down the line, clone those, and you getmore and more blockages.

18. Joshua Weiss says:

the clowns are disappearing again!

19. FaRo says:

Freeing the clowns!

20. FaRo says:

How many moves do you need to free two of the three prison cells?

21. Jason Kvasnak says:

The question was how to free the three prisoners from the prison.
The answer was fill all the space outside the prison with prisoners.
Now the prison is empty.

22. Cheefoo says:

Greenland used to be a nice country until the infinite clowns came and used up all their two gasolines.

23. OhDannyBoy512 says:

I love this explanation! So very clever!

24. Michael S says:

And even if you had infinite turns, you'd still have to free the last clone… but there's no space… so you'd have to free up the area around it, which is just like freeing the original prison. So you'd have to free an infinite number of prisons, from the first to the infinitieth, in reverse order.
And by that point, I suppose the entire board would be empty except for the one in the corner, since all the clones outside the prison would have to be in some "infinite" zone outside of the problem.

25. CNC Cocoa says:

Isn't it impossible to fill in the top right square of the infinite grid?

26. Aleksa Janic says:

#ClownsLifeMatters

27. Quanxiang Loo says:

This explanation is well done, but for those who are unsatisfied with some of the mysterious hand-waving and infinity arguments, don't worry. Using a similar idea, the proof can still be done by arguing for an upper bound of an n-by-n sized chessboard, and showing that for all n, the total area enclosed in the n-by-n chessboard is always less than 4.

28. Vincent says:

I don't like clowns. Specially if the they are clones. Damn cloned clowns!

29. //TNTz [-] says:

This is like the chain-reaction game

30. Spelling Mitsake says:

So, damn, amazing! I hate most of maths but when it's explained like this I'd love to know more. Extremely helpful.

31. Pratham Tandon says:

It feels like the proof mentioned in the video does not respect original rules of the game. The initial prison has 1,1/2 and 1/2. Now, one of the 1/2's splits into 2 quarters. But that disallows further splitting of the other 1/2, since for it to split, both its upper and right cells must be free! Hence, the 3rd row sum in the video cannot start with 1/4 :/

32. Random Guy says:

I don't get the series thing. You're adding a positive value to a total an infinite number of times. So shouldn't you end up with infinity?

33. VibratorDefibrilator says:

Euclid would be proud of this proof! Pure, finite logic!

34. Shaikh Mullah-ud-Din says:

♥ I am positively in love

35. scottmuck says:

Why do the "weights" of the clones have to reduce by half when cloned? That's critical to the proof but seems like an arbitrary assumption.

36. tromboneJTS says:

what is with saying "clowns" again and again. STOP clowning around.

37. Rishabh Dhiman says:

I saw this question first time in Arthur Engel's Problem Solving Strategies (Chapter 1), the Q was slightly different though, very slight.

38. Sondre Roos says:

Well done on question 6! Very impressive

39. hu ash says:

make 1 move,wait a minute,make another move,wait half a minute and so on
game finished in 2 minutes

40. SciGeoHistory says:

The answer is no, the Danish tried to populate Greenland and it didn't work at all.

41. MM93 says:

Not too sure ?

42. Chiranjeev Singh says:

If the rules say that no cloning allowed if the cell above or/and the cell to the right, then how is this [ 4:45 ] allowed?

43. 程天晞 says:

Numberphile, after seeing your, well, 2nd video I think, doesn't that prove that the original problem is unsolvable? There are only 2 clones on the first row or column after all, so the greenland is less than that of the prison! Are there any explanations to this?

44. Shahbaz Sheikh says:

Brilliant solution, explained brilliantly.

45. Ryan Roberson says:

i constructed a scenario where just the two clones trying to make room for the one clone will fill an indefinitely large square leaving only their original spots and an indefinitely long strip are clear, so this should definitely be impossible

46. Ryan Roberson says:

if you can prove that (proving something impossible) is impossible, then that is a contradiction since you just proved something impossible

47. Jordan says:

евала звезда

48. Rainy Chain says:

I'm a PC, I don't like infinity in mathematics (@,@")

49. Cellkist says:

Conway checkers is taking me on quite a journey

50. Michael Richardson says:

Should be called Freeing the CLOWNS. Much more fun.

51. Awwkaw says:

I hav problems with the weighing methog. I would expect the square (1;1) to have the weight 1/2 since it gets weight 1/4 from two different squares. This would allow a much higher weight outside the prison than inside.

52. Fareed Al-Bandar says:

If moving each clone takes a seconds (where a=the number assigned to the square the clone is on) won't the game finish in 4 seconds?

53. Simo Vihinen says:

Perhaps a simpler proof that you can't win would be this: the last move supposedly fills two holes but leaves one behind, but you can't fill that hole because you need two to make a move possible.

54. asaf attia says:

What a ridicules proof, i bet the original proof was difrent than just saying you cant doan infinat amout of moves.

55. Mantıkçı Mantık says:

Very elegant. Thank you very much. But now the new question: Can you free the clones in infinitely many moves? (With a one last move may be!) No! Because on the first row and the first column you can have at most two clones!

56. Sammy Thompson says:

A deity can solve the game. You just need sufficient 'power' over reality to do so. The game can be solved when time changes into eternity. The reality of eternity will instantaneously free the clones then and forevermore.

57. Jasper Boquiren says:

That's not a pebble

58. Anonymous71475 says:

Just wondering, but… is it possible to make it so that only one chip is left on the prison and that chip is on one of the "1/2" circle? Can't seem to do it :/

59. pro reviewer says:

Not to be rude or anything but I think I figured out that it is possible it would just take a really long time to do

60. Beachdude67 says:

In reality, the sum does wind up equaling 2. The sum should equal 1.99999 repeating, which actually equals 2. So this problem should be solvable.

61. IshaPls says:

2S = S + 2

2S – S = 2

S – (S/2) = 1

Slap that bad boy in the old ti89 and save yourself the watch

62. Looking In With Victor B says:

I don't get it… I tried it and won… So how did I do the impossible?

63. creeps Langford says:

Greenland is part of Denmark

64. Ra bert says:

Star Wars: the clown wars

65. Venkatesh babu says:

A inverse geometric sequence is always convergent . The limiting function is always less than the whole. So they are not greater or equal to escape.

66. raspy32official says:

How about not cloning clowns in the first place?

67. Ural Damasis says:

I think this this is officially the most horrendus accent in the world.

68. miles redmon says:

If you had the ability to play it for an infinitely long period of time could you then solve it?

69. Kevin Rose says:

I don't understand why these clowns are in prison. What country is this?

70. tochoXK3 says:

It is also impossible to make sure that only one clone that is NOT the one at the bottom left remains in the prison. If you're truly interested in the proof, answer this comment

71. Pedro Folloni Pesserl says:

7:07 I FINALLY UNDERSTOOD WHY 1+1/2+1/4+1/8+… IS 2! THANK YOU!

72. jimmyhackers says:

9:27 not sure that bit of maths/algebra is right

73. K1naku5ana3R1ka says:

Enough clowning around.

74. Kirito Kirigaya says:

Very elegant solution.

75. Zgembo121 says:

im here mostly for clown comments

76. Grey Squirrel says:

It would still not be possible to get all the clones out, even with infinite moves.

This is because when you make a move you cannot change the number of clones on the lowest row, or the leftmost column. So a maximum of two squares each on the lowest row and the left most column are occupied, with the others being empty.

Therefore you have to leave a non-zero amount of space in greenland empty, so the space the clones can occupy in greenland is less than 2.

77. Akshay Dhivare says:

I loved it

78. SUSHAE says:

poor greenland…

79. Føxtrøt says:

"lets assign these squares numbers, and our numbers for the boxes are going to be 'a' and 'b'."

80. Anthony Helms says:

Wouldn't you do 1/3 before you do 1/4? Where does the forth piece go?

81. thewebspinner says:

It's also worth mentioning that the first column and row can never be completely filled, in fact they'll be empty except for one stuck in the second row on the first column and one in the second column on the first row (or just one clone in it's original position in the bottom left corner) and one clone at the farthest end you've reached in both the first column and row. You don't really need to go any further because as stated in the video as soon as one square remains empty you know the sum cannot add up to a whole number. A fraction will always be left behind in the prison no matter how many moves you make.

If this still isn't clear: When starting you have three pieces all three HAVE to make a clone diagonally but only 2 can move vertically and 2 horizontally. Since none of the clones can jump down or to the right you cannot have more than 2 clones on the first row and column.
Just wanted to highlight this since in the video it doesn't demonstrate this and leads you to believe that all squares could be filled in an infinite game which simply isn't possible.

82. KanalDerGutenSache says:

They are actually called "echoes" in the new Smash Bros.

83. Indie Fedora Gamer says:

So basically the only way to free the clow… sorry clones is to do it as a super task.

84. Darshan Shah says:

Moral of the story: Don't play games with mathematicians!

85. Pirate's Piggy says:

The first number I learned… oh that one is too long to even fit in a comment section or a video

86. Maighe_TV says:

Explanation: we want to prove the game is impossible.
Step 1) conspiracy theories on the total sum of the square
Step 2) discovering the deep algorithm inside the game
Step 3) say "the game is infinite, so it must be impossible"

87. Ximo Mi Agenda Infantil says:

What a beautiful solution. Bravo!

88. Tony Ennis says:

89. agarRoyale 2002 says:

How are you going to put a clone in the top right corner?

90. MrChet407 says:

That's trippy

91. lithium191 says:

Stop clowning around

92. KiryambaBy nononono says:

Я вообще нихуя блять не понял

93. Vaklin Petkov says:

You make me proud to be a bulgarian.

94. Travis Terry says:

Jester from Critical Role?! 🤗

95. Craig B says:

Insane Clone Posse!

96. Super Gamer 2336 says:

This channel is Vsauce, but with maths, which makes me happy beyond reason

97. fartwrangler says:

Heh, "pebbles". After "stones" from backgammon?
Or, you could just call them "checkers", as we do in my country. 🙂

98. Yves Nyfeler Ph.D. says:

The very first and crucial assumption a=2b seems completely arbitrary to me and a bit of a stretch. Surely this has been rigorously proven somewhere but that step did not make much sense to me, given how many different ways there are to fill up your board in practice.

99. Mustavo Gaia says:

Why does the clone of 1 is 1/2? Shouldnt it be 1?

100. Ryan P says:

FREE THE CLOWNS