Modular Knights

450 /www/nrich/html/content/03/01/15plus1/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> A 'modular knight' moves on circular chess board made from concentric circles divided into sectors. As a default, the board is split into 5 sectors with 2 concentric tracks and the knight can move 3 steps forward (in any direction) followed by 1 step to the side (in either direction), as shown in the interactivity below. The middle and edge of the board are joined so that when the knight moves over the outside edge of the circular board it re-enters in the same sector on the inside of the track (and vice versa). Start the interactivity below by clicking on the +. The brown squares represent the squares the knight has visited and the peach squares the possible destinations on the next move. To begin with, understand why all of the peach squares are possibilities. Then, can you make the knight visit every square once and only once and return to its starting point? <mdo:flash height="362" width="650"><param value="/content/03/01/15plus1/dynamic_chess_board_V4.swf" name="movie" ></param><param value="6" name="flashplayerversion" ></param></mdo:flash> Suppose there are p sectors and q concentric tracks and a knight's move is a steps in one direction and b steps in the other direction. Find conditions on the numbers p, q, a and b under which it is possible for the knight to visit every square and return to its starting point. Note that on the interactivity you can change the size of the track, the direction and number of steps the knight can move forward, and the number of steps it can move to the side. </mdoxml> <?xml version="1.0" encoding="ISO-8859-1"?> <mdoxml version="1.0"> This problem was a toughnut for a while before we received solutions from several students. Max Whittaker from Culford School, Jose, Carlos , Dan and Sam from Ickford Combined School, Issy and Audrey, also from Ickford combined school, all sent in their methods of solution in certain cases. This involved starting in a certain place and moving around the board systematically. Alex from Darell Primary School gave an explicit solution: First you need to have these settings: knight moves round: 2 in/out: 1 board sectors: 3 tracks: 3 Here is what you should do in the exact order, there may be other ways, but here is what I did. Bottom/middle right Inner left Top right Inner bottom right Far left Bottom right Then finally, Far/middle left Ben from Kenny School gave a great description of his mathematical thinking about this analysis of this problem which we like very much: it really highlights the creative side of mathematical problem solving. Ben uses the notion of coprime numbers (two whole numbers which share no common factor except 1) to make some inroad into the analysis of the problem . Ben cleverly represented the board as a rectangle with opposite sides identified, which makes visualising, and therefore analysing, the situation much easier. If the grid is $p$ by $q$ and the knight can move $a$ across and $b$ down then Ben suggests that the journey is possible if at least 2 of the following conditions are true <ol> <li>$a (\mbox{ mod } p)$ is coprime to $p$</li> <li>$b (\mbox{ mod } q)$ is coprime to $q$</li> <li>$b (\mbox{ mod } p)$ is coprime to $p$</li> <li>$a (\mbox{ mod } q)$ is coprime to $q$</li> </ol> <div class="editorial">You can <a href="/content/03/01/15plus1/Ben%20solution.pdf">read Ben's analysis</a> here.</div> <div> </div> This is a great start, but not quite the whole story as some grids with coprime values are not coverable, although many are. <mdo:image height="221" width="597" alt="" src="solutionsSteve.png"></mdo:image> To make a little more progress, imagine that the grid has $p$ rows and $q$ columns, with $p, q$ coprime and $q> p$. Suppose also that the knight moves $a$ across and $b$ down each time, where $a$ is coprime to $p$ and $b$ coprime to $q$. Then each of the first $q$ squares will land on a different column of the grid ; therefore the knight will not have revisited any squares yet, although it will have revisited $q-p$ rows. If $q-p$ is coprime to $p$ then this procedure will fill the whole grid. If not, then the knight will need to change direction at some point to have a chance of filling the grid without hitting another square more than one. To proceed note that if the knight always moves in the same direction it positions will be of the form $(na \mbox{ mod p} , nb \mbox{ mod q}) $. For a location to be repeated, for some $n$ and $m$ we would have $$ (na \mbox{ mod p} , nb \mbox{ mod q}) =(ma \mbox{ mod p} , mb \mbox{ mod q}) $$ For this to hold we would have $$ ((n-m)a \mbox{ mod p} , (n-m)b \mbox{ mod q}) =(0, 0) $$ Since $a$ and $b$ are coprime to $p$ and $q$ respectively this implies that $(n-m) = Np$ and $(n-m)= Mq$ for some non-zero integers $N$ and $M$. Since $p$ and $q$ were chosen to be coprime and different we must have $N = Rq$ and $M=Rp$ for some other non-zero integer $R$. Thus $(n-m)$ is a non-zero multiple of $pq$. Since there are only $pq$ squares in the grid we see that the knight cannot have revisited any squares. </mdoxml> <mdoxml version="1.0">Keep a record of the set-ups when it is impossible for the knight to visit every square (that is the four numbers: of sectors, of tracks and of moves the knight makes in the two directions). What can you say about these set-ups? Consider common factors of pairs of these numbers. It may help to consider first the result for one track with p sectors and a knight's move of a steps forward and zero steps to the side for different values of p and a. What joint property of p and a determines whether or not the knight can visit every square? Try the standard knights move of two steps one way and one step the other way on different boards and see if the knight can visit every square. Now what happens if the knight moves two steps in each direction? A convenient notation is to let (x,y) => (x+s, y+t) denote a move of s steps clockwise and t steps outwards. As there are p sectors around the circuit the knight returns to the same sector after p steps clockwise which calls for taking x and x+s modulo p. Similarly as there are q tracks which the knight moves round cyclically, and the knight returns to the same track after taking q steps outwards, so take y and y+t modulo q. </mdoxml> <mdoxml version="1.0"> This problem was a toughnut for a while before we received solutions from several students. Max Whittaker from Culford School, Jose, Carlos , Dan and Sam from Ickford Combined School, Issy and Audrey, also from Ickford combined school, all sent in their methods of solution in certain cases. This involved starting in a certain place and moving around the board systematically. Alex from Darell Primary School gave an explicit solution: First you need to have these settings: knight moves round: 2 in/out: 1 board sectors: 3 tracks: 3 Here is what you should do in the exact order, there may be other ways, but here is what I did. Bottom/middle right Inner left Top right Inner bottom right Far left Bottom right Then finally, Far/middle left Ben from Kenny School gave a great description of his mathematical thinking about this analysis of this problem which we like very much: it really highlights the creative side of mathematical problem solving. Ben uses the notion of coprime numbers (two whole numbers which share no common factor except 1) to make some inroad into the analysis of the problem . Ben cleverly represented the board as a rectangle with opposite sides identified, which makes visualising, and therefore analysing, the situation much easier. If the grid is $p$ by $q$ and the knight can move $a$ across and $b$ down then Ben suggests that the journey is possible if at least 2 of the following conditions are true <ol> <li>$a (\mbox{ mod } p)$ is coprime to $p$</li> <li>$b (\mbox{ mod } q)$ is coprime to $q$</li> <li>$b (\mbox{ mod } p)$ is coprime to $p$</li> <li>$a (\mbox{ mod } q)$ is coprime to $q$</li> </ol> <div class="editorial">You can <a href="/content/03/01/15plus1/Ben%20solution.pdf">read Ben's analysis</a> here.</div> <div> </div> This is a great start, but not quite the whole story as some grids with coprime values are not coverable, although many are. <mdo:image height="221" width="597" alt="" src="solutionsSteve.png"></mdo:image> To make a little more progress, imagine that the grid has $p$ rows and $q$ columns, with $p, q$ coprime and $q> p$. Suppose also that the knight moves $a$ across and $b$ down each time, where $a$ is coprime to $p$ and $b$ coprime to $q$. Then each of the first $q$ squares will land on a different column of the grid ; therefore the knight will not have revisited any squares yet, although it will have revisited $q-p$ rows. If $q-p$ is coprime to $p$ then this procedure will fill the whole grid. If not, then the knight will need to change direction at some point to have a chance of filling the grid without hitting another square more than one. To proceed note that if the knight always moves in the same direction it positions will be of the form $(na \mbox{ mod p} , nb \mbox{ mod q}) $. For a location to be repeated, for some $n$ and $m$ we would have $$ (na \mbox{ mod p} , nb \mbox{ mod q}) =(ma \mbox{ mod p} , mb \mbox{ mod q}) $$ For this to hold we would have $$ ((n-m)a \mbox{ mod p} , (n-m)b \mbox{ mod q}) =(0, 0) $$ Since $a$ and $b$ are coprime to $p$ and $q$ respectively this implies that $(n-m) = Np$ and $(n-m)= Mq$ for some non-zero integers $N$ and $M$. Since $p$ and $q$ were chosen to be coprime and different we must have $N = Rq$ and $M=Rp$ for some other non-zero integer $R$. Thus $(n-m)$ is a non-zero multiple of $pq$. Since there are only $pq$ squares in the grid we see that the knight cannot have revisited any squares. </mdoxml> 2 5 0 0 0 0 1 Modular Knights Try to move the knight to visit each square once and return to the starting point. Move either 2 steps one way and one perpendicular (as in chess) or generalise to a steps one way and b the other. Numbers and the Number System Modulus arithmetic Decision Mathematics and Combinatorics Combinatorics Numbers and the Number System Factors and multiples Information and Communications Technology Interactivities