MaplePrimes - answers and comments on Question, how many routes in a 4x4 square

MaplePrimes - answers and comments on Question, how many routes in a 4x4 square http://www.mapleprimes.com/questions/142492-How-Many-Routes-In-A-4x4-Square en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Wed, 10 Jun 2026 07:14:16 GMT Wed, 10 Jun 2026 07:14:16 GMT The latest answers and comments added to the Question, how many routes in a 4x4 square http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - answers and comments on Question, how many routes in a 4x4 square http://www.mapleprimes.com/questions/142492-How-Many-Routes-In-A-4x4-Square Procedure http://www.mapleprimes.com/questions/142492-How-Many-Routes-In-A-4x4-Square?ref=Feed:MaplePrimes:how many routes in a 4x4 square:Comments#answer142513 <p>The procedure <strong>Routes</strong> finds all the routes with a travel length of <strong>n</strong> in a<strong> N</strong>x<strong>N</strong> square and the number of such routes.</p> <p><strong>Routes:=proc(N::posint, n::nonnegint)</strong></p> <p><strong>local L, Rule; global T;</strong></p> <p><strong>if n>=N^2 then T:=[]: print(0);   else</strong></p> <p><strong>L:=[seq(seq([[i, j]], j=1..N), i=1..N)];</strong></p> <p><strong> </strong></p> <p><strong>Rule:=proc(K)  # Continuation of the route by 1 step</strong></p> <p><strong>local S, k, M, r, p;</strong></p> <p><strong>S:=[ ]; k:=nops(K[1]);</strong></p> <p><strong>for r in K do</strong></p> <p><strong>M:=[ ]:</strong></p> <p><strong>if r[k]=[1, 1] then for p in [[1, 2], [2, 2], [2, 1]] do  # Bottom left corner </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k]=[1, N] then for p in [[1, N-1], [2, N-1], [2, N]] do  # Top left corner </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k]=[N, N] then for p in [[N-1, N], [N-1, N-1], [N, N-1]] do  # Top right corner </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k]=[N, 1] then for p in [[N-1, 1], [N-1, 2], [N, 2]] do  # Bottom right corner </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k,1]=1 and r[k,2]<>1 and r[k,2]<>N then for p in [[1, r[k,2]-1], [2, r[k,2]-1], [2, r[k,2]-1], [2, r[k,2]+1], [1, r[k,2]+1]] do                      # Left side </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k,2]=N and r[k,1]<>1 and r[k,1]<>N then for p in [[r[k,1]-1, N], [r[k,1]-1, N-1], [r[k,1], N-1], [r[k,1]+1, N-1], [r[k,1]+1, N]] do                    # Top side </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k,1]=N and r[k,2]<>1 and r[k,2]<>N then for p in [[N, r[k,2]+1], [N-1, r[k,2]+1], [N-1, r[k,2]], [N-1, r[k,2]-1], [N, r[k,2]-1]] do                    # Right side </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k,2]=1 and r[k,1]<>1 and r[k,1]<>N then for p in [[r[k,1]-1, 1], [r[k,1]-1, 2], [r[k,1], 2], [r[k,1]+1, 2], [r[k,1]+1, 1]] do                    # Bottom side </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k,1]<>1 and r[k,1]<>N and r[k,2]<>1 and r[k,2]<>N then for p in [[r[k,1]-1, r[k,2]-1], [r[k,1]-1, r[k,2]], [r[k,1]-1, r[k,2]+1], [r[k,1], r[k,2]+1], [r[k,1]+1, r[k,2]+1], [r[k,1]+1, r[k,2]], [r[k,1]+1, r[k,2]-1], [r[k,1], r[k,2]-1]] do  # Inside</strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>od;</strong></p> <p><strong>S;</strong></p> <p><strong>end proc;</strong></p> <p><strong> </strong></p> <p><strong>T:=(Rule@@n)(L);  # List of all the routes</strong></p> <p><strong>nops(T);  # Number of all the routes</strong></p> <p><strong>fi;</strong></p> <p><strong> </strong></p> <p><strong>end proc;</strong></p> <p><strong><br></strong></p> <p>Example:</p> <p><strong>Routes(4, 6); </strong></p> <p><strong>T[10000]; T[20000]; T[60000];</strong></p> <p><strong><img src="http://i024.radikal.ru/1301/7a/06b7be13839e.jpg" alt="" width="640" height="148"></strong></p> <p> </p> <p>The procedure <strong>Routes</strong> finds all the routes with a travel length of <strong>n</strong> in a<strong> N</strong>x<strong>N</strong> square and the number of such routes.</p> <p><strong>Routes:=proc(N::posint, n::nonnegint)</strong></p> <p><strong>local L, Rule; global T;</strong></p> <p><strong>if n>=N^2 then T:=[]: print(0);   else</strong></p> <p><strong>L:=[seq(seq([[i, j]], j=1..N), i=1..N)];</strong></p> <p><strong> </strong></p> <p><strong>Rule:=proc(K)  # Continuation of the route by 1 step</strong></p> <p><strong>local S, k, M, r, p;</strong></p> <p><strong>S:=[ ]; k:=nops(K[1]);</strong></p> <p><strong>for r in K do</strong></p> <p><strong>M:=[ ]:</strong></p> <p><strong>if r[k]=[1, 1] then for p in [[1, 2], [2, 2], [2, 1]] do  # Bottom left corner </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k]=[1, N] then for p in [[1, N-1], [2, N-1], [2, N]] do  # Top left corner </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k]=[N, N] then for p in [[N-1, N], [N-1, N-1], [N, N-1]] do  # Top right corner </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k]=[N, 1] then for p in [[N-1, 1], [N-1, 2], [N, 2]] do  # Bottom right corner </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k,1]=1 and r[k,2]<>1 and r[k,2]<>N then for p in [[1, r[k,2]-1], [2, r[k,2]-1], [2, r[k,2]-1], [2, r[k,2]+1], [1, r[k,2]+1]] do                      # Left side </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k,2]=N and r[k,1]<>1 and r[k,1]<>N then for p in [[r[k,1]-1, N], [r[k,1]-1, N-1], [r[k,1], N-1], [r[k,1]+1, N-1], [r[k,1]+1, N]] do                    # Top side </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k,1]=N and r[k,2]<>1 and r[k,2]<>N then for p in [[N, r[k,2]+1], [N-1, r[k,2]+1], [N-1, r[k,2]], [N-1, r[k,2]-1], [N, r[k,2]-1]] do                    # Right side </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k,2]=1 and r[k,1]<>1 and r[k,1]<>N then for p in [[r[k,1]-1, 1], [r[k,1]-1, 2], [r[k,1], 2], [r[k,1]+1, 2], [r[k,1]+1, 1]] do                    # Bottom side </strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>if r[k,1]<>1 and r[k,1]<>N and r[k,2]<>1 and r[k,2]<>N then for p in [[r[k,1]-1, r[k,2]-1], [r[k,1]-1, r[k,2]], [r[k,1]-1, r[k,2]+1], [r[k,1], r[k,2]+1], [r[k,1]+1, r[k,2]+1], [r[k,1]+1, r[k,2]], [r[k,1]+1, r[k,2]-1], [r[k,1], r[k,2]-1]] do  # Inside</strong></p> <p><strong>if convert([seq(r[i]<>p, i=1..k-1)], `and`) then M:=[op(M),[op(r), p]]: fi: </strong></p> <p><strong>od:</strong></p> <p><strong>S:=[op(S), op(M)]: fi:</strong></p> <p><strong>od;</strong></p> <p><strong>S;</strong></p> <p><strong>end proc;</strong></p> <p><strong> </strong></p> <p><strong>T:=(Rule@@n)(L);  # List of all the routes</strong></p> <p><strong>nops(T);  # Number of all the routes</strong></p> <p><strong>fi;</strong></p> <p><strong> </strong></p> <p><strong>end proc;</strong></p> <p><strong><br></strong></p> <p>Example:</p> <p><strong>Routes(4, 6); </strong></p> <p><strong>T[10000]; T[20000]; T[60000];</strong></p> <p><strong><img src="http://i024.radikal.ru/1301/7a/06b7be13839e.jpg" alt="" width="640" height="148"></strong></p> <p> </p> 142513 Tue, 22 Jan 2013 23:56:59 Z Kitonum Kitonum Improved version http://www.mapleprimes.com/questions/142492-How-Many-Routes-In-A-4x4-Square?ref=Feed:MaplePrimes:how many routes in a 4x4 square:Comments#answer142654 <p>By avoiding n^2 list building and using a few other refinements, I reduced the run time of Routes from about 20 seconds to 1/2 second. Here is the modified code</p> <pre>Routes:=proc(N::posint, n::nonnegint)<br><br>global T;<br>local i,j,L, Rule;<br><br>    if n>=N^2 then<br>        T:=[]:<br>        0;<br>    else<br>        Rule := proc(K)  # Continuation of the route by 1 step<br>        local S, k, r, rk, p, pts, j;<br><br>            S := table();<br>            j := 0;<br>            k := nops(K[1]);<br><br>            for r in K do<br>                # Assign pts the points to consider<br>                if r[k]=[1, 1] then<br>                    # Bottom left corner<br>                    pts := [[1, 2], [2, 2], [2, 1]];<br>                elif r[k]=[1, N] then<br>                    # Top left corner<br>                    pts := [[1, N-1], [2, N-1], [2, N]];<br>                elif r[k]=[N, N] then<br>                    # Top right corner<br>                    pts := [[N-1, N], [N-1, N-1], [N, N-1]];<br>                elif r[k]=[N, 1] then<br>                    # Bottom right corner<br>                    pts := [[N-1, 1], [N-1, 2], [N, 2]];<br>                elif r[k,1]=1 and r[k,2]<>1 and r[k,2]<>N then<br>                    # Left side<br>                    pts := [[1, r[k,2]-1], [2, r[k,2]-1], [2, r[k,2]-1], [2, r[k,2]+1], [1, r[k,2]+1]];<br>                elif r[k,2]=N and r[k,1]<>1 and r[k,1]<>N then<br>                    # Top side<br>                    pts := [[r[k,1]-1, N], [r[k,1]-1, N-1], [r[k,1], N-1], [r[k,1]+1, N-1], [r[k,1]+1, N]];<br>                elif r[k,1]=N and r[k,2]<>1 and r[k,2]<>N then<br>                    # Right side<br>                    pts := [[N, r[k,2]+1], [N-1, r[k,2]+1], [N-1, r[k,2]], [N-1, r[k,2]-1], [N, r[k,2]-1]];<br>                elif r[k,2]=1 and r[k,1]<>1 and r[k,1]<>N then<br>                    # Bottom side<br>                    pts := [[r[k,1]-1, 1], [r[k,1]-1, 2], [r[k,1], 2], [r[k,1]+1, 2], [r[k,1]+1, 1]];<br>                elif r[k,1]<>1 and r[k,1]<>N and r[k,2]<>1 and r[k,2]<>N then<br>                    # Inside<br>                    pts := [[r[k,1]-1, r[k,2]-1], [r[k,1]-1, r[k,2]], [r[k,1]-1, r[k,2]+1], [r[k,1], r[k,2]+1], [r[k,1]+1, r[k,2]+1], [r[k,1]+1, r[k,2]], [r[k,1]+1, r[k,2]-1], [r[k,1], r[k,2]-1]];<br>                fi;<br>                rk := r[1..k-1];<br>                j := j+1;<br>                S[j] := seq(`if`(member(p,rk)<br>                                 , NULL<br>                                 , [op(r),p]<br>                                ), p = pts);<br>            od;<br>            [seq(S[j], j=1..j)];<br>        end proc;<br><br>        L:=[seq(seq([[i, j]], j=1..N), i=1..N)];<br>        T:=(Rule@@n)(L);  # List of all the routes<br>        nops(T);  # Number of all the routes<br>    fi;<br>end proc:</pre> <p>By avoiding n^2 list building and using a few other refinements, I reduced the run time of Routes from about 20 seconds to 1/2 second. Here is the modified code</p> <pre>Routes:=proc(N::posint, n::nonnegint)<br><br>global T;<br>local i,j,L, Rule;<br><br>    if n>=N^2 then<br>        T:=[]:<br>        0;<br>    else<br>        Rule := proc(K)  # Continuation of the route by 1 step<br>        local S, k, r, rk, p, pts, j;<br><br>            S := table();<br>            j := 0;<br>            k := nops(K[1]);<br><br>            for r in K do<br>                # Assign pts the points to consider<br>                if r[k]=[1, 1] then<br>                    # Bottom left corner<br>                    pts := [[1, 2], [2, 2], [2, 1]];<br>                elif r[k]=[1, N] then<br>                    # Top left corner<br>                    pts := [[1, N-1], [2, N-1], [2, N]];<br>                elif r[k]=[N, N] then<br>                    # Top right corner<br>                    pts := [[N-1, N], [N-1, N-1], [N, N-1]];<br>                elif r[k]=[N, 1] then<br>                    # Bottom right corner<br>                    pts := [[N-1, 1], [N-1, 2], [N, 2]];<br>                elif r[k,1]=1 and r[k,2]<>1 and r[k,2]<>N then<br>                    # Left side<br>                    pts := [[1, r[k,2]-1], [2, r[k,2]-1], [2, r[k,2]-1], [2, r[k,2]+1], [1, r[k,2]+1]];<br>                elif r[k,2]=N and r[k,1]<>1 and r[k,1]<>N then<br>                    # Top side<br>                    pts := [[r[k,1]-1, N], [r[k,1]-1, N-1], [r[k,1], N-1], [r[k,1]+1, N-1], [r[k,1]+1, N]];<br>                elif r[k,1]=N and r[k,2]<>1 and r[k,2]<>N then<br>                    # Right side<br>                    pts := [[N, r[k,2]+1], [N-1, r[k,2]+1], [N-1, r[k,2]], [N-1, r[k,2]-1], [N, r[k,2]-1]];<br>                elif r[k,2]=1 and r[k,1]<>1 and r[k,1]<>N then<br>                    # Bottom side<br>                    pts := [[r[k,1]-1, 1], [r[k,1]-1, 2], [r[k,1], 2], [r[k,1]+1, 2], [r[k,1]+1, 1]];<br>                elif r[k,1]<>1 and r[k,1]<>N and r[k,2]<>1 and r[k,2]<>N then<br>                    # Inside<br>                    pts := [[r[k,1]-1, r[k,2]-1], [r[k,1]-1, r[k,2]], [r[k,1]-1, r[k,2]+1], [r[k,1], r[k,2]+1], [r[k,1]+1, r[k,2]+1], [r[k,1]+1, r[k,2]], [r[k,1]+1, r[k,2]-1], [r[k,1], r[k,2]-1]];<br>                fi;<br>                rk := r[1..k-1];<br>                j := j+1;<br>                S[j] := seq(`if`(member(p,rk)<br>                                 , NULL<br>                                 , [op(r),p]<br>                                ), p = pts);<br>            od;<br>            [seq(S[j], j=1..j)];<br>        end proc;<br><br>        L:=[seq(seq([[i, j]], j=1..N), i=1..N)];<br>        T:=(Rule@@n)(L);  # List of all the routes<br>        nops(T);  # Number of all the routes<br>    fi;<br>end proc:</pre> 142654 Sat, 26 Jan 2013 22:09:39 Z Joe Riel Joe Riel Flaw in Algorithm http://www.mapleprimes.com/questions/142492-How-Many-Routes-In-A-4x4-Square?ref=Feed:MaplePrimes:how many routes in a 4x4 square:Comments#answer142678 <p>This algorithm returns duplicates and also misses some paths.  To see that execute Routes(4,2) and inspect the first two results (T[1] and T[2]).  With duplicates removed, it returns only 82 paths for Route(4,1), the correct answer is easily computed to be 3*4 + 5*8 + 8*4 = 84.</p> <p>I believe the correct result, for Routes(4,6), is 68272. I computed that using a simpler approach, an extension to the Iterator package I've recently described.</p> <p><strong>Addendum</strong></p> <p>The original code for Routes give the correct answer for Routes(4,1), 84.  It is my modification of it that is flawed for that case. However, the original is incorrect with Routes(4,2), it has the duplicates I've mentioned. </p> <p><strong>Further Add</strong></p> <p>Actually, while the original code does return 84 for Routes(4,1), it returns only 82 unique routes.</p> <p><strong>Correction</strong></p> <p>The bug is in the points used for the left side; there is a duplicate.  With that fixed I get the same results as with the Iterator method. Here is the revised procedure</p> <pre>Routes:=proc(N::posint, n::nonnegint)<br><br>global T;<br>local i,j,L, Rule;<br><br>    if n>=N^2 then<br>        T:=[]:<br>        0;<br>    else<br>        Rule := proc(K)  # Continuation of the route by 1 step<br>        local S, k, r, rk, p, pts, j;<br><br>            S := table();<br>            j := 0;<br>            k := nops(K[1]);<br><br>            for r in K do<br>                # Assign pts the points to consider<br>                if r[k]=[1, 1] then<br>                    # Bottom left corner<br>                    pts := [[1, 2], [2, 2], [2, 1]];<br>                elif r[k]=[1, N] then<br>                    # Top left corner<br>                    pts := [[1, N-1], [2, N-1], [2, N]];<br>                elif r[k]=[N, N] then<br>                    # Top right corner<br>                    pts := [[N-1, N], [N-1, N-1], [N, N-1]];<br>                elif r[k]=[N, 1] then<br>                    # Bottom right corner<br>                    pts := [[N-1, 1], [N-1, 2], [N, 2]];<br>                elif r[k,1]=1 and r[k,2]<>1 and r[k,2]<>N then<br>                    # Left side<br>                    pts := [[1, r[k,2]-1], [2, r[k,2]-1], [2, r[k,2]], [2, r[k,2]+1], [1, r[k,2]+1]];<br>                elif r[k,2]=N and r[k,1]<>1 and r[k,1]<>N then<br>                    # Top side<br>                    pts := [[r[k,1]-1, N], [r[k,1]-1, N-1], [r[k,1], N-1], [r[k,1]+1, N-1], [r[k,1]+1, N]];<br>                elif r[k,1]=N and r[k,2]<>1 and r[k,2]<>N then<br>                    # Right side<br>                    pts := [[N, r[k,2]+1], [N-1, r[k,2]+1], [N-1, r[k,2]], [N-1, r[k,2]-1], [N, r[k,2]-1]];<br>                elif r[k,2]=1 and r[k,1]<>1 and r[k,1]<>N then<br>                    # Bottom side<br>                    pts := [[r[k,1]-1, 1], [r[k,1]-1, 2], [r[k,1], 2], [r[k,1]+1, 2], [r[k,1]+1, 1]];<br>                elif r[k,1]<>1 and r[k,1]<>N and r[k,2]<>1 and r[k,2]<>N then<br>                    # Inside<br>                    pts := [[r[k,1]-1, r[k,2]-1], [r[k,1]-1, r[k,2]], [r[k,1]-1, r[k,2]+1], [r[k,1], r[k,2]+1], [r[k,1]+1, r[k,2]+1], [r[k,1]+1, r[k,2]], [r[k,1]+1, r[k,2]-1], [r[k,1], r[k,2]-1]];<br>                fi;<br>                rk := r[1..k-1];<br>                j := j+1;<br>                S[j] := seq(`if`(member(p,rk)<br>                                 , NULL<br>                                 , [op(r),p]<br>                                ), p = pts);<br>            od;<br>            [seq(S[j], j=1..j)];<br>        end proc;<br><br>        L:=[seq(seq([[i, j]], j=1..N), i=1..N)];<br>        T:=(Rule@@n)(L);<br>        nops(T);  # Number of all the routes<br>    fi;<br>end proc:</pre> <p>This algorithm returns duplicates and also misses some paths.  To see that execute Routes(4,2) and inspect the first two results (T[1] and T[2]).  With duplicates removed, it returns only 82 paths for Route(4,1), the correct answer is easily computed to be 3*4 + 5*8 + 8*4 = 84.</p> <p>I believe the correct result, for Routes(4,6), is 68272. I computed that using a simpler approach, an extension to the Iterator package I've recently described.</p> <p><strong>Addendum</strong></p> <p>The original code for Routes give the correct answer for Routes(4,1), 84.  It is my modification of it that is flawed for that case. However, the original is incorrect with Routes(4,2), it has the duplicates I've mentioned. </p> <p><strong>Further Add</strong></p> <p>Actually, while the original code does return 84 for Routes(4,1), it returns only 82 unique routes.</p> <p><strong>Correction</strong></p> <p>The bug is in the points used for the left side; there is a duplicate.  With that fixed I get the same results as with the Iterator method. Here is the revised procedure</p> <pre>Routes:=proc(N::posint, n::nonnegint)<br><br>global T;<br>local i,j,L, Rule;<br><br>    if n>=N^2 then<br>        T:=[]:<br>        0;<br>    else<br>        Rule := proc(K)  # Continuation of the route by 1 step<br>        local S, k, r, rk, p, pts, j;<br><br>            S := table();<br>            j := 0;<br>            k := nops(K[1]);<br><br>            for r in K do<br>                # Assign pts the points to consider<br>                if r[k]=[1, 1] then<br>                    # Bottom left corner<br>                    pts := [[1, 2], [2, 2], [2, 1]];<br>                elif r[k]=[1, N] then<br>                    # Top left corner<br>                    pts := [[1, N-1], [2, N-1], [2, N]];<br>                elif r[k]=[N, N] then<br>                    # Top right corner<br>                    pts := [[N-1, N], [N-1, N-1], [N, N-1]];<br>                elif r[k]=[N, 1] then<br>                    # Bottom right corner<br>                    pts := [[N-1, 1], [N-1, 2], [N, 2]];<br>                elif r[k,1]=1 and r[k,2]<>1 and r[k,2]<>N then<br>                    # Left side<br>                    pts := [[1, r[k,2]-1], [2, r[k,2]-1], [2, r[k,2]], [2, r[k,2]+1], [1, r[k,2]+1]];<br>                elif r[k,2]=N and r[k,1]<>1 and r[k,1]<>N then<br>                    # Top side<br>                    pts := [[r[k,1]-1, N], [r[k,1]-1, N-1], [r[k,1], N-1], [r[k,1]+1, N-1], [r[k,1]+1, N]];<br>                elif r[k,1]=N and r[k,2]<>1 and r[k,2]<>N then<br>                    # Right side<br>                    pts := [[N, r[k,2]+1], [N-1, r[k,2]+1], [N-1, r[k,2]], [N-1, r[k,2]-1], [N, r[k,2]-1]];<br>                elif r[k,2]=1 and r[k,1]<>1 and r[k,1]<>N then<br>                    # Bottom side<br>                    pts := [[r[k,1]-1, 1], [r[k,1]-1, 2], [r[k,1], 2], [r[k,1]+1, 2], [r[k,1]+1, 1]];<br>                elif r[k,1]<>1 and r[k,1]<>N and r[k,2]<>1 and r[k,2]<>N then<br>                    # Inside<br>                    pts := [[r[k,1]-1, r[k,2]-1], [r[k,1]-1, r[k,2]], [r[k,1]-1, r[k,2]+1], [r[k,1], r[k,2]+1], [r[k,1]+1, r[k,2]+1], [r[k,1]+1, r[k,2]], [r[k,1]+1, r[k,2]-1], [r[k,1], r[k,2]-1]];<br>                fi;<br>                rk := r[1..k-1];<br>                j := j+1;<br>                S[j] := seq(`if`(member(p,rk)<br>                                 , NULL<br>                                 , [op(r),p]<br>                                ), p = pts);<br>            od;<br>            [seq(S[j], j=1..j)];<br>        end proc;<br><br>        L:=[seq(seq([[i, j]], j=1..N), i=1..N)];<br>        T:=(Rule@@n)(L);<br>        nops(T);  # Number of all the routes<br>    fi;<br>end proc:</pre> 142678 Sun, 27 Jan 2013 05:54:15 Z Joe Riel Joe Riel Generalization makes this much simpler http://www.mapleprimes.com/questions/142492-How-Many-Routes-In-A-4x4-Square?ref=Feed:MaplePrimes:how many routes in a 4x4 square:Comments#answer142762 <p>Sometimes---more often than I would expect---solving a more general problem is easier yet still gives a complete solution to the original problem. Such is the case here. The problem at hand is equivalent to finding all directed paths of a given length in a graph. What you call routes are formally called paths: a sequence of distinct vertices, each adjacent to the preceding.</p> <p>Once the data structure is defined as I have done in the procedure below, finding the paths is essentially a one-liner. There's no need to consider special cases like vertices on corners, sides, etc.</p> <p><strong>AllPaths:= proc(</strong><br><strong>     G::GraphTheory:-Graph</strong><br><strong>    ,maxlength::nonnegint:= GraphTheory:-NumberOfVertices(G)</strong><br><strong>)</strong><br><strong>#Returns a table R such that R[k] is a list of all directed paths of length k in G,</strong><br><strong>#where k ranges from 0 to maxlength. G can be directed or undirected.</strong><br><strong>#Each path is represented as a list of vertices.</strong><br><strong>#The algorithm is breadth-first search.</strong><br><strong>    uses GT= GraphTheory;</strong><br><strong>    local</strong><br><strong>         k  #current length </strong><br><strong>        ,p  #current path</strong><br><strong>        ,v  #current vertex</strong><br><strong>        ,V:= GT:-Vertices(G)  #constant</strong><br><strong>        ,E:= table(V =~ (`{}`@op)~ (GT:-Departures(G)))  #constant table of edges</strong><br><strong>        ,R:= table([0= `[]`~ (V)])  #Initialize with zero-length paths.</strong><br><strong>    ;</strong><br><strong>    for k to maxlength do</strong><br><strong>         #For each path of length k-1, make all possible 1-vx extensions.</strong><br><strong>        R[k]:= [seq](seq([p[], v], v in E[p[-1]] minus {p[]}), p in R[k-1])</strong><br><strong>    end do; </strong><br><strong>    eval(R) </strong><br><strong>end proc:</strong></p> <p>Now apply it to the problem at hand:<strong></strong></p> <p><strong>squares:= [seq](seq([i,j], i= 1..4), j= 1..4):</strong></p> <p>Name the vertices conveniently.</p> <p><strong>V:= (v-> cat(['a','b','c','d'][v[1]], v[2]))~ (squares);</strong></p> <p>V := [a1, b1, c1, d1, a2, b2, c2, d2, a3, b3, c3, d3, a4, b4, c4, d4]<strong></strong></p> <p><strong>Vsq:= table(V =~ squares):</strong></p> <p>Define the edges of the graph. Horizontal, vertical, and diagonal adjacency are all covered by simply saying that the infinity norm of the difference of the points is 1.</p> <p><strong>Sq44:= GraphTheory:-Graph(<br>   select(e-> LinearAlgebra:-Norm(<Vsq[e[1]] - Vsq[e[2]]>) = 1, combinat:-choose({V[]}, 2))<br>):</strong></p> <p>Generate the paths and time it.</p> <p><strong>CodeTools:-Usage(assign(Pths, AllPaths(Sq44, 6))):</strong></p> <p>memory used=11.31MiB, alloc change=4.27MiB, cpu time=219.00ms, real time=210.00ms<strong></strong></p> <p>Explore the results a bit.</p> <p><strong>nops(Pths[6]);</strong></p> <p>68272<strong></strong></p> <p>Get a well-distributed sample of the paths.</p> <p><strong>Samp:= [seq](Pths[6][2^k], k= 1..16);</strong></p> <p>Samp := [[a1, a2, a3, a4, b3, b2, c1], [a1, a2, a3, a4, b3, b2, c3], [a1, a2, a3, a4, b3, c2, b2], [a1, a2, a3, a4, b3, c3, c2], [a1, a2, a3, a4, b4, c3, c4], [a1, a2, a3, b2, b3, c3, d4], [a1, a2, a3, b2, c3, d3, c4], [a1, a2, a3, b4, c3, b3, c4], [a1, a2, b1, c2, c3, b4, a3], [a1, a2, b3, b4, a3, b2, c2], [a1, b1, c1, c2, d3, c3, b3], [a2, a1, b2, a3, b3, c2, d1], [a2, b3, c3, d2, c1, c2, b2], [a4, b4, c3, c2, b1, c1, b2], [b4, c3, d2, c1, b1, a2, a1], [d4, c3, d2, c2, b3, b4, a3]]</p> <p>Develop a few tools for visualizing the paths.</p> <p>To center a vertex in its square:<strong></strong></p> <p><strong>pt:= v-> Vsq[v] -~ 1/2:</strong></p> <p>Background to plot on:</p> <p><strong>grid:= plot(0, view= [0..4, 0..4], gridlines, scaling= constrained):</strong></p> <p><strong>pathplot:= p-><br>   plots:-display(<br>      grid <br>     ,plot(<br>         [[pt](p[1]), [pt](p[-1]), pt~(p)]<br>        ,style= [point, point, line], color= [green, red, yellow]<br>        ,symbol= diamond, symbolsize= 30<br>      )<br>   )<br>:</strong></p> <p>And finally here are the 16 sample paths:</p> <p><strong>plots:-display(Matrix(4,4, pathplot~ (Samp)));</strong></p> <p>(Sorry, I don't know how to upload an array of plots. If you execute the above code yourself, you will get a 4x4 array of plots---one for each of the sample paths.)<strong><br></strong></p> <p><strong><a href="/view.aspx?sf=142762/452535/PathPlot.gif"><img src="/view.aspx?sf=142762/452535/PathPlot.gif" alt="" width="281" height="398"></a></strong></p> <p>Sometimes---more often than I would expect---solving a more general problem is easier yet still gives a complete solution to the original problem. Such is the case here. The problem at hand is equivalent to finding all directed paths of a given length in a graph. What you call routes are formally called paths: a sequence of distinct vertices, each adjacent to the preceding.</p> <p>Once the data structure is defined as I have done in the procedure below, finding the paths is essentially a one-liner. There's no need to consider special cases like vertices on corners, sides, etc.</p> <p><strong>AllPaths:= proc(</strong><br><strong>     G::GraphTheory:-Graph</strong><br><strong>    ,maxlength::nonnegint:= GraphTheory:-NumberOfVertices(G)</strong><br><strong>)</strong><br><strong>#Returns a table R such that R[k] is a list of all directed paths of length k in G,</strong><br><strong>#where k ranges from 0 to maxlength. G can be directed or undirected.</strong><br><strong>#Each path is represented as a list of vertices.</strong><br><strong>#The algorithm is breadth-first search.</strong><br><strong>    uses GT= GraphTheory;</strong><br><strong>    local</strong><br><strong>         k  #current length </strong><br><strong>        ,p  #current path</strong><br><strong>        ,v  #current vertex</strong><br><strong>        ,V:= GT:-Vertices(G)  #constant</strong><br><strong>        ,E:= table(V =~ (`{}`@op)~ (GT:-Departures(G)))  #constant table of edges</strong><br><strong>        ,R:= table([0= `[]`~ (V)])  #Initialize with zero-length paths.</strong><br><strong>    ;</strong><br><strong>    for k to maxlength do</strong><br><strong>         #For each path of length k-1, make all possible 1-vx extensions.</strong><br><strong>        R[k]:= [seq](seq([p[], v], v in E[p[-1]] minus {p[]}), p in R[k-1])</strong><br><strong>    end do; </strong><br><strong>    eval(R) </strong><br><strong>end proc:</strong></p> <p>Now apply it to the problem at hand:<strong></strong></p> <p><strong>squares:= [seq](seq([i,j], i= 1..4), j= 1..4):</strong></p> <p>Name the vertices conveniently.</p> <p><strong>V:= (v-> cat(['a','b','c','d'][v[1]], v[2]))~ (squares);</strong></p> <p>V := [a1, b1, c1, d1, a2, b2, c2, d2, a3, b3, c3, d3, a4, b4, c4, d4]<strong></strong></p> <p><strong>Vsq:= table(V =~ squares):</strong></p> <p>Define the edges of the graph. Horizontal, vertical, and diagonal adjacency are all covered by simply saying that the infinity norm of the difference of the points is 1.</p> <p><strong>Sq44:= GraphTheory:-Graph(<br>   select(e-> LinearAlgebra:-Norm(<Vsq[e[1]] - Vsq[e[2]]>) = 1, combinat:-choose({V[]}, 2))<br>):</strong></p> <p>Generate the paths and time it.</p> <p><strong>CodeTools:-Usage(assign(Pths, AllPaths(Sq44, 6))):</strong></p> <p>memory used=11.31MiB, alloc change=4.27MiB, cpu time=219.00ms, real time=210.00ms<strong></strong></p> <p>Explore the results a bit.</p> <p><strong>nops(Pths[6]);</strong></p> <p>68272<strong></strong></p> <p>Get a well-distributed sample of the paths.</p> <p><strong>Samp:= [seq](Pths[6][2^k], k= 1..16);</strong></p> <p>Samp := [[a1, a2, a3, a4, b3, b2, c1], [a1, a2, a3, a4, b3, b2, c3], [a1, a2, a3, a4, b3, c2, b2], [a1, a2, a3, a4, b3, c3, c2], [a1, a2, a3, a4, b4, c3, c4], [a1, a2, a3, b2, b3, c3, d4], [a1, a2, a3, b2, c3, d3, c4], [a1, a2, a3, b4, c3, b3, c4], [a1, a2, b1, c2, c3, b4, a3], [a1, a2, b3, b4, a3, b2, c2], [a1, b1, c1, c2, d3, c3, b3], [a2, a1, b2, a3, b3, c2, d1], [a2, b3, c3, d2, c1, c2, b2], [a4, b4, c3, c2, b1, c1, b2], [b4, c3, d2, c1, b1, a2, a1], [d4, c3, d2, c2, b3, b4, a3]]</p> <p>Develop a few tools for visualizing the paths.</p> <p>To center a vertex in its square:<strong></strong></p> <p><strong>pt:= v-> Vsq[v] -~ 1/2:</strong></p> <p>Background to plot on:</p> <p><strong>grid:= plot(0, view= [0..4, 0..4], gridlines, scaling= constrained):</strong></p> <p><strong>pathplot:= p-><br>   plots:-display(<br>      grid <br>     ,plot(<br>         [[pt](p[1]), [pt](p[-1]), pt~(p)]<br>        ,style= [point, point, line], color= [green, red, yellow]<br>        ,symbol= diamond, symbolsize= 30<br>      )<br>   )<br>:</strong></p> <p>And finally here are the 16 sample paths:</p> <p><strong>plots:-display(Matrix(4,4, pathplot~ (Samp)));</strong></p> <p>(Sorry, I don't know how to upload an array of plots. If you execute the above code yourself, you will get a 4x4 array of plots---one for each of the sample paths.)<strong><br></strong></p> <p><strong><a href="/view.aspx?sf=142762/452535/PathPlot.gif"><img src="/view.aspx?sf=142762/452535/PathPlot.gif" alt="" width="281" height="398"></a></strong></p> 142762 Tue, 29 Jan 2013 03:07:57 Z Carl Love Carl Love Vote up http://www.mapleprimes.com/questions/142492-How-Many-Routes-In-A-4x4-Square?ref=Feed:MaplePrimes:how many routes in a 4x4 square:Comments#comment142514 <p>However, your comments are not enough detailed.</p> <p>However, your comments are not enough detailed.</p> 142514 Wed, 23 Jan 2013 00:26:06 Z Markiyan Hirnyk Markiyan Hirnyk More comments http://www.mapleprimes.com/questions/142492-How-Many-Routes-In-A-4x4-Square?ref=Feed:MaplePrimes:how many routes in a 4x4 square:Comments#comment142517 <p><a href="http://www.mapleprimes.com/questions/142492-How-Many-Routes-In-A-4x4-Square#comment142514">@Markiyan Hirnyk</a></p> <p>The subprocedure  <strong>Rule </strong> is of fundamental importance in procedure <strong> Routes</strong>. It allows you to extend the already created list of routes <strong> K</strong>   one more step. Applying  <strong>Rule </strong> to the list of all the vertices <strong> L </strong>, we get a list of all routes of length 1. Applying the procedure  <strong>Rule</strong>  to the previous list, we get all routes of length 2, and so on.</p> <p> </p> <p> </p> <p><a href="http://www.mapleprimes.com/questions/142492-How-Many-Routes-In-A-4x4-Square#comment142514">@Markiyan Hirnyk</a></p> <p>The subprocedure  <strong>Rule </strong> is of fundamental importance in procedure <strong> Routes</strong>. It allows you to extend the already created list of routes <strong> K</strong>   one more step. Applying  <strong>Rule </strong> to the list of all the vertices <strong> L </strong>, we get a list of all routes of length 1. Applying the procedure  <strong>Rule</strong>  to the previous list, we get all routes of length 2, and so on.</p> <p> </p> <p> </p> 142517 Wed, 23 Jan 2013 01:11:05 Z Kitonum Kitonum