MathZoom, Summer, 2014

Transcription

1 A one-dimensional bug starts at the origin and each minute moves either left or right exactly one unit. Suppose it makes there moves with equal likelihood. That is the probability of a move to the left is /2 and to the right is /2. What is the probability that after exactly 6 moves, the bug is back at the starting point? This is the type of problem we can explore here. The objective is to explore the relationship between bug movements on a line, in a plane, on the edges of a cube, etc. with certain polynomials Now compare each line of the triangular array with a polynomial of the form (l + r) n. So the top row corresponds to the polynomial (l + r) 0 =, the next row to (l + r) = l + r, the next is (l + r) 2 = ll + lr + rl + rr = l 2 + 2lr + r 2.

2 . Suppose the bug is on a cube. At each minute the bug crawls to an adjacent vertex, each time with equal likelihood. What is the probability that after exactly 8 minutes, the bug is back where it started? 2. Now our bug is a chess king in the plane. We ask questions like how many paths of length 6 are there that start at the origin and end at (4, 4)? 3. A bug starts crawling along the edges of a cube so that at each vertex it takes each edge with probability /3. What is the probability that after seven moves, the bug will have visited every vertex exactly once? 4. A bug in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? 5. A bug starts at the point (0, 0) of an xy-coordinate grid and them makes a sequence of six moves. Each move is unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of y = x is m, where m and N are relatively n prime positive integers. Find m + n. All the following problems except number 2 refer to a bijective function f : Z + Z Z such that d(f(n), f(n + )) 2 for all n and f(0) = (0, 0). Such a function is called a crawl. Here, Z + = {0,, 2, 3,...} is the set of nonnegative integers, and Z is the set of all integers. Think of f as describing the crawl of a bug, with elements of Z + as times. A turning point of a bug crawl is a number t such that the bug is crawling one direction during the minute before t and another direction during the minute afterwards.. A bug crawls around the plane at a uniform rate, one unit per minute. He starts at the origin at time 0 and crawls one unit to the right, arriving at (, 0), turns 90 left and crawls another unit to (, ), turns 90 left again, and crawls two units. He continues to make 90 left turns as shown in the figure. Let g(t) denote the position in the plane after t minutes, where t is an integer. Thus, for example, g(0) = (0, 0), g(6) = (, ), and g(6) = ( 2, 2). (The path of the bug establishes a one-to-one correspondence 2

3 between the non-negative integers and the integer lattice points of the plane.) (0, 0) (a) Where is the bug after exactly 2008 minutes? How many turns does the bug make during the first 2008 minutes of its crawl? (b) How many minutes does it take for the bug to get to the ordered pair (9, 99)? (c) Does there exist an integer t such that g(t) and g(t + 23) are exactly 7 units apart? If so, find the smallest such t. (d) Now suppose one bug leaves exactly 99 minutes after the other. What is the closest they ever get to each other? In other words, what is the smallest possible value of D(g(t), g(t+99)), t 0, where D represents the distance function? What is the smallest value of t for which this distance is achieved. (e) Now suppose one bug leaves exactly 00 minutes after the other. What is the closest they ever get to each other? In other words, what is the smallest possible value of D(g(t), g(t + 00)), t 0, where D represents the distance function? What is the smallest value of t for which this distance is achieved. (f) For how many integer values of t is g(t) exactly 5 units from the origin? If we drop the integer requirement and assume the bug travels at a uniform rate, how many times is it exactly 5 units from the origin? Find the largest and smallest integers t for which g(t) is 5 units from the origin. Find the largest and smallest real numbers for which g(t) is 5 units from the origin. 3

4 (g) Now suppose another bug leaves the origin exactly 7 minutes later and follows the same path at the same rate. At what time are the bugs first exactly 5 units apart? How many times during the first half hour are the bugs exactly 5 units apart? (0, 0) (h) Build the function g : Z + Z Z that describes the bugs crawl. (i) Build the function h : Z Z Z + that describes the time when the bug reaches each lattice point of the plane. (j) Find all the turning points. (k) Build the parametric functions that describe the x and y coordinates as a function of t. 4

5 2. This time the bug stays in the first quadrant. It crawls from the origin up one unit after one minute, over to (, ) after another minute, and down to (, 0) after three minutes. Each time it gets to an axis, it makes a right turn away from the origin, crawls one unit, then makes another right turn, staying in the first quadrant. (0, 0) (a) Where is the bug after exactly 2008 minutes? 5

6 (b) How many minutes does it take for the bug to get to the ordered pair (9, 99)? 3. A bug starts from the origin on the plane and crawls one unit upwards to (0, ) after one minute. During the second minute, it crawls two units to the right ending at (2, ). Then during the third minute, it crawls three units upward, arriving at (2, 4). It makes another right turn and crawls four units during the fourth minute. From here it continues to crawl n units during minute n and then makes a 90 turn either left or right. The bug continues this until after 6 minutes, it finds itself back at the origin. Its path does not intersect itself. What is the smallest possible area of the 6-gon traced out by its path? 4. This time our bug can make turns other than right turns. Consider the path shown below, again starting at the origin. Each unit segment between lattice points take exactly one minute to traverse and each diagonal segment of length 2 also takes one minute..... (0, 0) (a) Where is the bug after exactly 2008 minutes? (b) How many minutes does it take for the bug to get to the ordered pair (9, 99)? (c) Find the smallest t for which the bugs position g(t) after t minutes is at least 00 units away from the origin. For this value of t, how many turns has the bug made? 6

7 5. This time our bug moves around based on where it is. Let and A = {(m, n) m > 2/3 and m n < 2m + }, B = {(m, n) n 2m + and n m/2}, C = {(m, n) n m/2 and n m }. As before g(t) is the position of the bug at time t. Now we are given that g(0) = (0, 0). Also, when the bug belongs to set A at time t, it moves up one unit at t +. If g(t) = (a, b) belongs to B, then g(t + ) = (a, b ), and if g(t) = (a, b) belongs to C, then g(t + ) = (a, b). (a) Where is the bug after exactly 000 minutes? (b) How many minutes does it take for the bug to get to the ordered pair (2, 22)? (c) Find the smallest t for which the bugs position g(t) after t minutes is at least 00 units away from the origin. For this value of t, how many turns has the bug made? 6. Here s yet another crawl. This time, we ll give the bug lots of steps of length 2. 7

8 (0, 0) g(2) = ( 2, 0).. (a) How long does it take the bug to reach the point (20, 0)? (b) Where is the bug after 200 minutes? (c) Build a square around the first n rings, find the area A in square units of the square, and use this to estimate the location of the bug after A minutes. 8

9 7. For each bug crawl above, let D(t) denote the total distance the bug has crawled at time t. Let L = lim. t Does this limit always exist? Find L for each of the crawls above. t inf D(t) 8. Project. A bug crawls around the plane at a rate that gets it to a new integer lattice point every minute. He starts at the origin at time 0 and crawls one unit upward, arriving at (0, ), turns and crawls directly to (, 0) arriving at time 2, turns 90 left and crawls 2 units to the point (0, ) arriving at time 3. Let g(t) denote the position in the plane after t minutes, where t is an integer. Thus, for example, g(0) = (0, 0), g(4) = (, 0), and g(2) = (2, 0). (a) How long does it take the bug to reach the point (20, 0)? (b) Where is the bug after 200 minutes? (c) Build the function g and its inverse function. 2 g(4) = (, 0).. g(2) = (2, 0) 9

10 2 g(4) = (, 0).. g(2) = (2, 0) 9. (Mathcounts 202, National Team, 9) A lattice polygon in the plane is a polygon P all of whose vertices are integer lattice points. A rectangular lattice octagon(rlo) is a lattice octagon in the plane such that each pair of consecutive edges are perpendicular. Suppose n is a positive integer. (a) Is there a rectangular lattice octagon O with sides of lengths n, n +, n + 2,..., n + 7? (b) In case n =, what are the RLO s with the greatest and least areas. (c) Generalize the result of (b). 0. (A on the 20 Putnam Competition) Define a growing spiral in the plane to be a sequence of points with integer coordinates P 0 = (0, 0), P,..., P n such that n 2 and: The directed line segments P 0 P, P P 2,..., P n P n are in the successive coordinate directions east (for P 0 P ), north, west, south, east, etc. The lengths of these line segments are positive and strictly increasing. [Picture omitted.] How many of the points (x, y) with integer coordinates 0 x 20, 0 y 20 cannot be the last point, P n of any growing spiral? References and further reading. 0

11 References [] discusses the locations of prime numbers in the crawl. [2] discusses integer spirals. [3] provides problems and solution for the Putnam Exams,