Investigating the Classic Crossed Ladders Puzzle

Today's blog post is a complete working MATLAB program investigating the crossed ladders problem. Download a copy of the program via the link at the end. Publish it again with the publish command or the publish editor tab.

Contents

The model

Explore the interaction of eight variables satisfying five nonlinear equations.

The eight variables are:

  • $a$, length of one ladder.
  • $b$, length of the other ladder.
  • $c$, height of the point where they cross.
  • $u$, width of one base triangle.
  • $v$, width of the other base triangle.
  • $w$, width of the alley.
  • $x$, height of the point where one ladder meets the wall.
  • $y$, height of the point where the other ladder meets the other wall.

The five equations are:

$$ a^2 = x^2 + w^2 $$

$$ b^2 = y^2 + w^2 $$

$$ \frac{x}{w} = \frac{c}{v} $$

$$ \frac{y}{w} = \frac{c}{u} $$

$$ w = u + v $$

ladders_app

The initial values provide the smallest solution that is all integers. Are there any other integer solutions within the range of this app?

function ladders_app
    % The Classic Crossed Ladders Puzzle.
    % See Cleve's Corner, Feb. 29, 2016.
    % <https://blogs.mathworks.com/cleve/2016/02/29/the-classic-crossed-ladders-puzzle>

    % Starting values are a minimal sum integer solution.
    w0 = 56;
    x0 = 105;
    y0 = 42;
    w = w0;
    x = x0;
    y = y0;
    c = x*y/(x+y);
    u = c*w/y;
    v = c*w/x;

    % Starting toggles have both labels off.
    letters = false;
    numbers = false;

    initialize_figure;

letters

Snapshot with letters = true and a wider valley.

numbers

Snapshot of the mirror image of the previous values with numbers = true. Four of the eight values are still integers.

initialize_figure

    function initialize_figure
        % Size of the square view
        h = 160;
        clf
        shg
        set(gcf,'menubar','none', ...
            'numbertitle','off', ...
            'name','Crossed Ladders', ...
            'windowbuttondownfcn',@down, ...
            'windowbuttonupfcn',@up)
        axis([-h/8 7/8*h -h/8 7/8*h])
        axis square
        lines
        buttons
        labels
    end %initialize_figure

motion

Called repeatedly as the mouse moves.

    function motion(varargin)
        % WindowsButtonMotionFunction
        [p,q] = read_mouse;
        u = c*w/y;
        v = c*w/x;
        % Which control point?
        if q < c/2
            % Drag the alley width horizonally.
            w = p;
        elseif p < u/2
            % Drag the ladder against the left wall vertically.
            x = q;
        elseif p > w-v/2
            % Drag both the width and the other ladder.
            w = p;
            y = q;
        else
            % Drag the crossing point, creating lots of action.
            c = q;
            u = p;
            v = w-u;
            x = c*w/v;
            y = c*w/u;
        end
        c = x*y/(x+y);
        lines
    end % motion

lines

Play dot-to-dot.

    function lines
        cla
        ms = 3*get(gcf,'defaultlinemarkersize');  % Large dots
        u = c*w/y;

        % Five markers at the vertices.
        line([0 0],[0 0],'Marker','.','MarkerSize',ms)
        line([w w],[0 0],'Marker','.','MarkerSize',ms,'Color','k')
        line([0 0],[x x],'Marker','.','MarkerSize',ms,'Color','k')
        line([w w],[y y],'Marker','.','MarkerSize',ms,'Color','k')
        line([u u],[c c],'Marker','.','MarkerSize',ms,'Color','k')

        % Connect the markers with six lines.
        line([0 w],[0 0])
        line([0 0],[0 x])
        line([w w],[0 y])
        line([0 w],[x 0])
        line([w 0],[y 0])
        line([u u],[c 0],'LineStyle','-.')
        box on
    end % lines

down

    function down(varargin)
        % Called at the start of mouse movement.
        % Activate the motion function.
        cla
        set(gcf,'windowbuttonmotionfcn',@motion)
    end % down

up

    function up(varargin)
        % Called at the end of mouse movement.
        % Deactivate motion function.
        set(gcf,'windowbuttonmotionfcn',[])
        set(gcf,'windowbuttondownfcn',@down)
        labels
    end % up

buttons

    function buttons
        % Two toggles and three pushbuttons.
        posit = [.88 .64 .10 .06];
        delta = [  0 .08   0   0];

        % letters
        uicontrol('style','toggle', ...
            'units','normalized', ...
            'position',posit, ...
            'string','letters', ...
            'value',letters, ...
            'callback',@letters_cb);

        % numbers
        uicontrol('style','toggle', ...
            'units','normalized', ...
            'position',posit-delta, ...
            'string','numbers', ...
            'value',numbers, ...
            'callback',@numbers_cb);

        % mirror
        uicontrol('style','pushbutton', ...
            'units','normalized', ...
            'position',posit-2*delta, ...
            'string','mirrror', ...
            'callback',@mirror_cb);

        % reset
        uicontrol('style','pushbutton', ...
            'units','normalized', ...
            'position',posit-3*delta, ...
            'string','reset', ...
            'callback',@reset_cb);

        % close
        uicontrol('style','pushbutton', ...
            'units','normalized', ...
            'position',posit-4*delta, ...
            'string','close', ...
            'callback','close(gcf)');
    end % buttons

labels

    function labels
        % Label lines with either letters or numbers.
        u = c*w/y;
        v = c*w/x;
        lines
        if letters
            f = '%s';
            xs = 'x  ';
            ys = '  y';
            ws = ' w ';
            as = '  a';
            bs = 'b  ';
            us = ' u ';
            vs = ' v ';
            cs = 'c  ';
            top = 'Drag a black dot.';
        elseif numbers
            % This is the only place values of a and b are needed.
            a = sqrt(x^2 + w^2);
            b = sqrt(y^2 + w^2);
            z = [a b c u v w x y];
            if all(abs(z-round(z)) < .001)
                f = '%.0f';
            else
                f = '%.1f';
            end
            if sum(z) < 50 || sum(z) > 5000
                scream
            end
            xs = x;
            ys = y;
            ws = w;
            as = a;
            bs = b;
            us = u;
            vs = v;
            cs = c;
            top = ['sum = ' sprintf(f,sum(z))];
        else
            top = 'Drag a black dot.';
        end
        if letters || numbers
            text(0,x/2,sprintf(f,xs),'horiz','right')
            text(w,y/2,sprintf(f,ys),'horiz','left')
            text(w/2,-3,sprintf(f,ws),'horiz','center')
            text(w/2,x/2,sprintf(f,as),'horiz','left')
            text(w/2,y/2,sprintf(f,bs),'horiz','right')
            text(u/2,3,sprintf(f,us),'horiz','center')
            text(w-v/2,3,sprintf(f,vs),'horiz','center')
            text(u,c/2,sprintf(f,cs),'horiz','right')
        end
        title(top)
     end % labels

letters_cb

    function letters_cb(varargin)
        % Called when letters button is toggled.
        letters = get(findobj(gcf,'string','letters'),'value');
        % Make sure numbers is false.
        numbers = false;
        set(findobj(gcf,'string','numbers'),'value',numbers)
        labels
    end % letter_cb

numbers_cb

    function numbers_cb(varargin)
        % Called when numbers button is toggled.
        numbers = get(findobj(gcf,'string','numbers'),'value');
        % Make sure letters is false.
        letters = false;
        set(findobj(gcf,'string','letters'),'value',letters)
        labels
    end % numbers_cb

mirror_cb

    function mirror_cb(varargin)
        % Called when mirror button is pushed.
        % Interchange x and y.
        t = x;
        x = y;
        y = t;
        % c = x*y/(x+y) is unchanged
        lines
        labels
    end % mirror_cb

reset_cb

    function reset_cb(varargin)
        % Called when reset button is pushed.
        % Restore initial values.
        w = w0;
        x = x0;
        y = y0;
        c = x*y/(x+y);
        u = c*w/y;
        v = c*w/x;
        lines
        labels
    end % reset_cb

read_mouse

    function [p,q] = read_mouse
        % Current horizontal and vertical coordinates of the mouse.
        pq = get(gca,'currentpoint');
        p = pq(1,1);
        q = pq(1,2);
    end % read_mouse
end % ladders_app




Published with MATLAB® R2016a

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