Campamento Verano OMPR 2011
June, 3, 2011
Invariants and Coloring Jiacheng Feng Part I We first start off with a few warm-up problems to get a hang of the topic: S 1. Several stones are placed on an infinite (in both directions) strip of squares. As long as there are at least two stones on a single square, you may pick up two such stones, and then move one to the preceding square and one to the following square. Is it possible to return to the starting configuration after a finite sequence of such moves?
Hint: Label the strip with consecutive integers, and let nidenote the label of the square containing stone #i. Consider X=∑ni2 after each move. S 2. (Problem Solving Strategies) Al writes the numbers 1, 2, 3, ‌, 4n+1,4n+2 on the board. Each turn, he picks two numbers, a and b, erases them, and write |a-b|. Show that an odd number will remain in the end. Hint: After each move, the parity of the sum of all numbers are invariant. In both problems, we see that using invariants is a key method to solving some difficult combinatorics problem. Although finding the invariant is difficult, the result will provide new insight into the problem and sometimes even trivialize it The idea to solving invariant problems is:
If there is a repetition, look for what does not change (Or at least changes with a pattern)
Here are a few things that you should think about doing invariance problems: “1 Algebraic expressions: Given a set of values, look at their differences, their sum, the sum of their squares, or occasionally their product. If you are working with integers, try looking at these values modulo n. (Usually n should be a small prime power.) 2 Corners and edges: For grid-based problems, consider any shapes formed. How many boundary edges do they have? How many corners? 3 Inversions: If you are permuting a sequence of numbers, consider the number of inversions that is, the number of pairs (i; j) such that i and j are listed in the wrong order. Both the absolute number of inversions and its parity are useful.
4 Integers and rational numbers: Can you find a positive integer that keeps decreasing? Or does the denominator of a rational number keep decreasing? 5 Colorings: Color all the squares in a grid with two or more colors. Usually the chessboard pattern is a good choice, but other patterns are also sometimes useful. Consider squares of each color separately. 6 Symmetries: Can you ensure that after each step, a figure is symmetrical in some way? Perhaps you can logically pair up objects, and two paired objects are always in the same state? This is especially useful for game-theory type problems.[1]�
Now, I will provide 6 problems on invariance to solve, such that each problem employs one of the 6 methods stated above.
P 1 (well-known) This is the famous Seven Bridges of KĂśnigsberg:
Find a path through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time. P 2 (Kvant) The first six terms of a sequence are 0, 1, 2, 3, 4, 5. Each subsequent term is the last digit of the sum of the six previous terms. In other words, the seventh term is 5; the eighth term is 0, etc. Can the subsequence 1, 3, 5, 7, 0occur anywhere? How about 1, 0, 1, 0, 1, 1 P 3 (USAMO 2011 part) An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn consists of subtracting an integer m from each of the integers at two neighboring vertices and adding 2m to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount m and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove if the game can be won, there is exactly one vertex at which the game can be won. P 4 (Baltic way 2006) A child wrote three numbers 5,6,7. Every minute, he deleted two numbers, say a and b. Then he wrote two numbers again, they were 0.6a+0.8b and0.8a – 0.6b. Is it possible that he got three numbers 2,6,10 at some point? P 5 (Problem Solving Strategies [2]) The number 1,2,3...,n are written in a row. It is permitted to swap any two numbers. If 2007 such operations are performed, is it possible that the final arrangement of numbers coincide with the original?
P6 (Problem Solving Strategies [2]) Solve: (x2-3x+3) 2-3(x2-3x+3)+3=x
Part II Today, we will also look at the coloring/tiling method, which is one of the most commonly used invariance method. “Key ideas: color the board in some way that gives some constraints or invariants. Often, this involves marking a certain subset of the board. Looking at parities and other modulo is a good idea. Sometimes you may have to use more than one coloring schemes simultaneously to solve a problem. Finally, don't forget that some problem have two (related) parts - proving a constraint and constructing an example. [3]” Here is the classic coloring problem: S1 How many ways can we use 1x2 dominos to tile a 8x8 board with the opposite corners removed? Hint: Color checkerboard style and see what colors are the squares each domino covers.
P1 Can a 4x11 rectangle be filled with L shaped pieces, its rotation and reflection?
P 2 (PSS) Can a 10x10 rectangle be filled with straight pieces and its rotation? (Try to find 2 coloring proofs)
P 3 (PSS) On a 4 x n board, is it possible to find a closed knight tour? That is the knight will start on a square, travel through all the other squares exactly once, and return back to the original square. P 4 (Centro American Math Olympiad 2010) board, and is moved according to the A token is placed in one square of a following rules:
In each turn, the token can be moved to a square sharing a side with the one currently occupied. The token cannot be placed in a square that has already been occupied.
Any two consecutive moves cannot have the same direction.
The game ends when the token cannot be moved. Determine the values of and for which, by placing the token in some square, all the squares of the board will have been occupied in the end of the game. P 5 (Netherland 2008) There are 2012 cards laid in a row on the floor, each with an integer written on the faceup side. Albert and Betty can take turns picking up one card from either the leftmost side or the rightmost side of the row. The person who accumulates the cards with the largest sum wins. Show that if Albert goes first, he has a non-losing strategy. P 6 (Gong) What is the minimum number of colored squares on a (2n) x (2n) board such that each individual square has an adjacent (not including diagonally connected) square that is colored?
Fun Fact: An 8 x 8 chessboard can actually be tiled by 1 x 2 dominos in 24•9012=12 988 816 ways. Thanks References: [1] http://web.mit.edu/yufeiz/www/wc08/invariants.pdf [2] A. Engel, Problem Solving Strategies, 1998, Springfield Science LLC (chapters 1 and 2) [3] http://web.mit.edu/yufeiz/www/olympiad/tiling.pdf
Giveaway Hints Part I P1:
Draw a complete graph which lands/islands are points and bridges are lines. Consider then the paths to go “in” and “out” of each point
P2:
a) modulo 2, show some sequence is invariant.
b) Let Xi=an+ 2an+1+ 3an+2+ 4an+3+ 5an+4+ 6an+5, where ai is the ith number in the sequence. Show it is invariant modulo 5 P3:
Label vertices a, b, c, d, e. Let fi be the value of vertex i. Let ga = 1, gb =2, gc =3, gd =4, ge =0. Sum of fi∙gi stays invariant mod 5.
P4: P5:
Show sum of the squares of the three numbers are invariant. Call a j>k, but j is at the left of k an inversion. Each swap of neighboring integers causes +1 or -1 inversions. However, each swap is the same as swapping neighboring integers an odd amount of times
P6:
Let f(x)= x2-3x+3; we need to find roots to f(f(x))=x. If a is a root of f(x)=x, then a is a root of (f(x))=x Part II
P1.
Consider columns of alternating Black White coloring. After each move, how many of each color will the L-piece color? How many pieces are there? How many of each color are there?
P2
Try this configuration, and notice what the straight piece covers. 1 2 3 4 1 …
P3
2 3 4 1 2 …
3 4 1 2 3 …
4 1 2 3 4 …
1 2 3 4 1 …
… … … … … …
Note that in this coloring, a knight at 1 must go to 3. But there are equal colors of 1 and 3. If there is a closed tour, there must be more 3 than 1. Think about why. 1 3 4 2
2 4 3 1
1 3 4 2
2 4 3 1
1 3 4 2
… … … …
P4:
Checkerboard coloring. Note that every two moves must be the same as going diagonally (Black to Black). Now consider the “in” paths and “out” paths for the endpoints as well as points close to the starting point (cells pointed by the arrows) 1 2 1 …
2 1 2 …
1 2 1 …
2 1 2 …
1 2 1 …
… … … …
P5
Color alternative Black and White, with first card black. Then the ending card is white. If Albert picks black card, what choices do Betty have?
P6
Color checkerboard style. Consider each color square separately. How many black squares needs to be marked such that all white squares have an adjacent square marked (look at black squares)? Then, what is the minimum amount of black squares needed to be marked such that all white squares have an adjacent square marked (look at white squares)?