Elementary Techniques used in the IMO (International Mathematical Olympiad) - Invariants

This post continues from both: Part 1 and Part 2

Part 3 3


Applications

The first example is of a game. We will see how to apply the invariance principle.

Example 1: On a chessboard A and B play by turns to place black and white knights, respectively. Either player loses when he places a knight on a square attacked by a knight of the other colour or there are no free squares to place the knight on. If A starts, who has a winning strategy?

Solution: For those chess noobs like me, remark that a knight can only attack the 8 8 squares shown in the figure below:

tagtext tagtext

Intuitively, if each player can place a knight such that the squares that can be attacked by both knights do not overlap, then since a chessboard has an even number of cells, Player B can win, by in what people call, mirroring the first player.

Clearly a knight on a black square can only attack a knight on a white square and vice versa. So the colours of the squares that the knight attacks is invariant. Recall that a chessboard is a 8×8 8\times 8 , so if we "fold" the board along the middle, the 2 2 "flaps" would be of opposing colour. For instance, let ai,j a_{i,j} be the colour of the cell in the ithith row and jth jth column of the chessboard. Then we have that a4,2=a4,7 a_{4,2} = a_{4,7} .

The above observation is key for us to see that the strategy for B is to do the same thing as A but symmetric to the center of the board. Basically, notice that if A can place a knight without being attacked by B's knights, the B can always place a "symmetric" knight without being attacked by A's knight due to the colour argument. Now given the way that knights attack, A can never place a knight that attacks the square where B played. Thus A loses in the end. □

To further train the skills of the reader in solving game-like problems (because we would soon be moving to colouring and other invariance, and I would select the IMO problems for demonstration), we will attempt one more example, this problem was in my training notes, and according to our trainer, was from Bulgaria 2001.

Example 2 Alice and Bob play by turns to write ones and zeroes in a list, from left to right. The game ends when each has written 2001 numbers. When the game ends the sequence of 0,1 0 ,1 is interpreted in base 2 2 . Alice wins if that number can be written as a sum of two perfect squares and if otherwise, Bob wins. Who has a winning strategy?

Solution: We want a invariant that somehow involves squares. Well, preliminarily, just plain squares would remind me of (mod4) \pmod 4 . So ultimately one should think of Fermat's Christmas Theorem. Let's see how this property translates to a winning strategy.

Now suppose the number is 11000 1100 \ldots 0 where there is an even number of 0 0 , what can you observe? (Hint: consider (mod4) \pmod 4 ) Yup, exactly! Since 4m4m can be written as a sum of 2 2 squares iff m m can, we can perfectly well ignore the zeroes at the end. And notice that 11 11 in base 2 2 cannot be written as a sum of 2 2 squares, so B wins! Generalising this argument, at any moment A writes a 1 1 , B just copies As A's move. In this way, if we ignore the zeores, the final number would end in 11 11 which is congruent to 3(mod4) 3 \pmod 4 which by Fermat's Christmas Theorem, cannot be written as a sum of 2 2 squares, so B wins.

Now, remember that A A is smart. She wants to avoid losing to B B in this way.

Can you complete the argument? Post your completed proof in the comments.


Announcement: I am extremely sorry but since I will be going overseas and the wifi there is unsteady, I will not be posting any more of these daily instalments until Friday, Singapore time.

Note by Anqi Li
5 years, 11 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

simo questions...

Wei Jie Tan - 5 years, 11 months ago

Log in to reply

Obviously

Anqi Li - 5 years, 11 months ago

Log in to reply

simo?

John Muradeli - 5 years, 2 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...