Elegant, not Elephant
by Ankoganit, Jul 31, 2016, 4:20 AM
"Any good idea can be stated in fifty words or less." - Stanislaw Ulam
The purpose of this post is collect some illuminating proofs where a small insight trivializes the seemingly difficult problem. The proofs need not be exactly one-line long; the general (and flexible) criteria for inclusion is
Now have a look at some of the one-liners I have encountered so far:
Alice and Bob are playing a game with the set 
. They make alternating moves, with Alice going first. At each move, one has to choose a number among the set which has not been chosen by anyone before. If at some point, a player has three numbers adding up to 
among those (s)he has chosen, (s)he wins. Prove that Alice can force a draw at least.
A Limp King is a chess piece that can move one square in any direction except for northeast and southwest.
![[asy]import graph; size(2cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-5.937420718816065,xmax=7.136575052854118,ymin=-3.4221141649048596,ymax=4.358012684989431;
draw((0.,0.)--(0.,2.),EndArrow(6)); draw((0.,0.)--(0.,-2.),EndArrow(6)); draw((0.,0.)--(2.,0.),EndArrow(6)); draw((0.,0.)--(-2.,0.),EndArrow(6)); label("N",(0,2.5)); label("S",(0,-2.5)); label("W",(-2.5,0)); label("E",(2.5,0));
dot((0.,0.),linewidth(3.pt)+ds);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]](//latex.artofproblemsolving.com/8/e/d/8eda3609e9d305a87f7c90c0052a231f4e6f3c64.png)
In a standard 
chessboard, some squares are to be destroyed. To cross the board, the Limp King has to start on any undestroyed square on the North side of the board, and make valid moves passing through undestroyed squares, and end up on an undestroyed square on the South edge of the board. How many ways are there to destroy exactly 
squares, such that the Limp King cannot cross the board?
Let 
be a positive integer greater than 
. Prove that there exists an irrational number 
such that 
is a rational number.
An isosceles right-angled triangle shaped billiards table has three pockets at the vertices. A ball starts moving from one of the pockets adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if it falls into a pocket then it is not the pocket whence it started.
The game of two-move chess follows the usual rules of chess with one exception: each player has to make two consecutive moves at a time. Prove that White (who goes first) has a non-losing strategy.
I will try to update this list as often as possible. You are welcome to suggest other one-line proofs you know to add to this post.
Sources:
1. An AoPS topic here, 2. A post on Puzzling.SE here, 3. own, here, 4. Iran TST 2007, 5. Mathematical Miniatures, Svetoslav Savchev and Titu Andreescu.
The purpose of this post is collect some illuminating proofs where a small insight trivializes the seemingly difficult problem. The proofs need not be exactly one-line long; the general (and flexible) criteria for inclusion is
- The solution must not be obvious; it must be unexpected and surprising.
- It should preferably demonstrate the interdisciplinary interconnections in mathematics.
- And of course, as noted above, the central idea should be describable in fifty words or less.
Now have a look at some of the one-liners I have encountered so far:
Alice and Bob are playing a game with the set 
. They make alternating moves, with Alice going first. At each move, one has to choose a number among the set which has not been chosen by anyone before. If at some point, a player has three numbers adding up to 
among those (s)he has chosen, (s)he wins. Prove that Alice can force a draw at least.
magic square. Then it is equivalent to Tic-Tac-Toe, and it is well known that the first player always has a strategy to avoid losing. 
A Limp King is a chess piece that can move one square in any direction except for northeast and southwest.
![[asy]import graph; size(2cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-5.937420718816065,xmax=7.136575052854118,ymin=-3.4221141649048596,ymax=4.358012684989431;
draw((0.,0.)--(0.,2.),EndArrow(6)); draw((0.,0.)--(0.,-2.),EndArrow(6)); draw((0.,0.)--(2.,0.),EndArrow(6)); draw((0.,0.)--(-2.,0.),EndArrow(6)); label("N",(0,2.5)); label("S",(0,-2.5)); label("W",(-2.5,0)); label("E",(2.5,0));
dot((0.,0.),linewidth(3.pt)+ds);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]](http://latex.artofproblemsolving.com/8/e/d/8eda3609e9d305a87f7c90c0052a231f4e6f3c64.png)
chessboard, some squares are to be destroyed. To cross the board, the Limp King has to start on any undestroyed square on the North side of the board, and make valid moves passing through undestroyed squares, and end up on an undestroyed square on the South edge of the board. How many ways are there to destroy exactly 
squares, such that the Limp King cannot cross the board?
cases. 
Let 
be a positive integer greater than 
. Prove that there exists an irrational number 
such that 
is a rational number.
.
,
.
such that 
and 
And since 
w
.
An isosceles right-angled triangle shaped billiards table has three pockets at the vertices. A ball starts moving from one of the pockets adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if it falls into a pocket then it is not the pocket whence it started.
,
and 
,
The game of two-move chess follows the usual rules of chess with one exception: each player has to make two consecutive moves at a time. Prove that White (who goes first) has a non-losing strategy.I will try to update this list as often as possible. You are welcome to suggest other one-line proofs you know to add to this post.
Sources:
1. An AoPS topic here, 2. A post on Puzzling.SE here, 3. own, here, 4. Iran TST 2007, 5. Mathematical Miniatures, Svetoslav Savchev and Titu Andreescu.
This post has been edited 1 time. Last edited by Ankoganit, Jul 31, 2016, 4:27 AM
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