Henry Ernest Dudeney/Puzzles and Curious Problems/75 - A Question of Transport/Working

Working for Puzzles and Curious Problems by Henry Ernest Dudeney: $75$ -- A Question of Transport

The system of simultaneous equations in matrix form:

$\begin {pmatrix}

1 &  0 &  0 &  0 & -20 &   0 &   0 &   0 &  0 \\
0 &  1 &  0 &  0 &   0 &  -4 &   0 &   0 &  0 \\
1 & -1 &  0 &  0 &  20 & -20 &   0 &   0 &  0 \\

-1 & 0 & 1 & 0 & 4 & 0 & -4 & 0 & 0 \\

0 & -1 &  1 &  0 &   0 &  20 & -20 &   0 &  0 \\
0 & -1 &  0 &  1 &   0 &   4 &   0 &  -4 &  0 \\
0 &  0 &  1 & -1 &   0 &   0 &  20 & -20 &  0 \\
0 &  0 &  1 &  0 &   0 &   0 &  -4 &   0 &  4 \\
0 &  0 &  0 &  1 &   0 &   0 &   0 & -20 & 20 \\

\end {pmatrix} \begin {pmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ t_1 \\ t_2 \\ t_3 \\ t_4 \\ t_5 \end {pmatrix} = \begin {pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 20 \\ 20 \end {pmatrix}$

when converted to echelon form, gives:

$\begin {pmatrix}

1 &  0 &  0 &  0 & -20 &   0 &    0 &    0 &    0 \\
0 &  1 &  0 &  0 &   0 &  -4 &    0 &    0 &    0 \\
0 &  0 &  1 &  0 & -16 &   0 &   -4 &    0 &    0 \\
0 &  0 &  0 &  1 &   0 &   0 &    0 &   -4 &    0 \\
0 &  0 &  0 &  0 &   1 &   1 &   -1 &    0 &    0 \\
0 &  0 &  0 &  0 &   0 &   1 & -5/2 &  3/2 &    0 \\
0 &  0 &  0 &  0 &   0 &   0 &    1 &   -1 & -1/6 \\
0 &  0 &  0 &  0 &   0 &   0 &    0 &    1 &  5/6 \\
0 &  0 &  0 &  0 &   0 &   0 &    0 &    0 &    1 \\ 

\end {pmatrix} \begin {pmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ t_1 \\ t_2 \\ t_3 \\ t_4 \\ t_5 \end {pmatrix} = \begin {pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ -5/6 \\ 25/6 \\ 13/5 \end {pmatrix}$

Proof

\(\ds \)

\(\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 20 & -20 & 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 & 4 & 0 & -4 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 & 20 & -20 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 & 4 & 0 & -4 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 & 20 & -20 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 4 & 20 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -20 & 20 & 20 \\ \end {array} }\)

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 40 & -20 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 & 20 & -20 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 & 4 & 0 & -4 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 & 20 & -20 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 4 & 20 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -20 & 20 & 20 \\ \end {array} }\)

$r_3 \to r_3 - r_1$, $r_4 \to r_4 + r_1$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 40 & -24 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 16 & -20 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 & 20 & -20 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 4 & 20 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -20 & 20 & 20 \\ \end {array} }\)

$r_3 \to r_3 + r_2$, $r_5 \to r_5 + r_2$, $r_6 \to r_6 + r_2$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 16 & -20 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 40 & -24 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 & 20 & -20 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 4 & 20 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -20 & 20 & 20 \\ \end {array} }\)

rearranging

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 16 & 16 & -16 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 40 & -24 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & -1 & 16 & 0 & 24 & -20 & 0 & 0 \\ 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -20 & 20 & 20 \\ \end {array} }\)

$r_4 \to r_4 - r_3$, $r_7 \to r_7 - r_3$, $r_8 \to r_8 - r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 16 & 16 & -16 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 40 & -24 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 16 & 0 & 24 & -20 & 0 & 0 \\ 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -20 & 20 & 20 \\ \end {array} }\)

rearranging

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 16 & 16 & -16 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 40 & -24 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 16 & 0 & 24 & -24 & 0 & 0 \\ 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 20 & 20 \\ \end {array} }\)

$r_7 \to r_7 + r_4$, $r_9 \to r_9 - r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 40 & -24 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 16 & 0 & 24 & -24 & 0 & 0 \\ 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 20 & 20 \\ \end {array} }\)

$r_5 \to r_5 / 16$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -64 & 40 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -16 & 40 & -24 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -16 & 16 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 20 & 20 \\ \end {array} }\)

$r_6 \to r_6 - 40 r_5$, $r_7 \to r_7 - 16 r_5$, $r_8 \to r_8 - 16 r_5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -16 & 40 & -24 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -64 & 40 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -16 & 16 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 20 & 20 \\ \end {array} }\)

rearranging

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -5/2 & 3/2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -64 & 40 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -16 & 16 & 0 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 20 & 20 \\ \end {array} }\)

$r_6 \to -r_6 / 16$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -5/2 & 3/2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -120 & 96 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -24 & 48 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 20 & 20 \\ \end {array} }\)

$r_7 \to r_7 + 64 r_6$, $r_8 \to r_8 + 16 r_6$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -5/2 & 3/2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -24 & 24 & 4 & 20 \\ 0 & 0 & 0 & 0 & 0 & 0 & -120 & 96 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 20 & 20 \\ \end {array} }\)

rearranging

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -5/2 & 3/2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1/6 & -5/6 \\ 0 & 0 & 0 & 0 & 0 & 0 & -5 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 20 & 20 \\ \end {array} }\)

$r_7 \to -r_7 / 24$, $r_8 \to -r_8 / 24$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -5/2 & 3/2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1/6 & -5/6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -5/6 & -25/6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 20 & 20 \\ \end {array} }\)

$r_8 \to r_8 + 5 r_7$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -5/2 & 3/2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1/6 & -5/6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 5/6 & 25/6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 20 & 20 \\ \end {array} }\)

$r_8 \to -r_8$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -5/2 & 3/2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1/6 & -5/6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 5/6 & 25/6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 100/3 & 260/3 \\ \end {array} }\)

$r_9 \to r_9 + 16 r_8$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccccccc{{|}}c} 1 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -16 & 0 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -5/2 & 3/2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1/6 & -5/6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 5/6 & 25/6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 13/5 \\ \end {array} }\)

$r_9 \to r_9 \div 100/3$

$\blacksquare$