Henry Ernest Dudeney/Puzzles and Curious Problems/134 - The Bag of Nuts/Working

Working for Puzzles and Curious Problems by Henry Ernest Dudeney: $134$ -- The Bag of Nuts

The simultaneous equations in matrix form:

$\begin {pmatrix}

1 &  1 &  1 &  1 &  1 \\
1 &  1 &  0 &  0 &  0 \\
0 &  1 &  1 &  0 &  0 \\
0 &  0 &  1 &  1 &  0 \\
0 &  0 &  0 &  1 &  1 \\

\end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ d \\ e \end {pmatrix} = \begin {pmatrix} 100 \\ 52 \\ 43 \\ 34 \\ 30 \end {pmatrix}$

when converted to reduced echelon form, gives:

$\begin {pmatrix}

1 &  0 &  0 &  0 &  0 \\
0 &  1 &  0 &  0 &  0 \\
0 &  0 &  1 &  0 &  0 \\
0 &  0 &  0 &  1 &  0 \\
0 &  0 &  0 &  0 &  1 \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ d \\ e \end {pmatrix} = \begin {pmatrix} 27 \\ 25 \\ 18 \\ 16 \\ 14 \end {pmatrix}$

Proof

\(\ds \)

\(\)

\(\ds \paren {\begin {array} {ccccc{{|}}c} 1 & 1 & 0 & 0 & 0 & 52 \\ 0 & 1 & 1 & 0 & 0 & 43 \\ 0 & 0 & 1 & 1 & 0 & 34 \\ 0 & 0 & 0 & 1 & 1 & 30 \\ 1 & 1 & 1 & 1 & 1 & 100 \\ \end {array} }\)

rearranging for convenience

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccc{{|}}c} 1 & 1 & 0 & 0 & 0 & 52 \\ 0 & 1 & 1 & 0 & 0 & 43 \\ 0 & 0 & 1 & 1 & 0 & 34 \\ 0 & 0 & 0 & 1 & 1 & 30 \\ 0 & 0 & 1 & 1 & 1 & 48 \\ \end {array} }\)

$r_5 \to r_5 - r_1$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccc{{|}}c} 1 & 1 & 0 & 0 & 0 & 52 \\ 0 & 1 & 1 & 0 & 0 & 43 \\ 0 & 0 & 1 & 1 & 0 & 34 \\ 0 & 0 & 0 & 1 & 1 & 30 \\ 0 & 0 & 0 & 0 & 1 & 14 \\ \end {array} }\)

$r_5 \to r_5 - r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccc{{|}}c} 1 & 1 & 0 & 0 & 0 & 52 \\ 0 & 1 & 1 & 0 & 0 & 43 \\ 0 & 0 & 1 & 1 & 0 & 34 \\ 0 & 0 & 0 & 1 & 0 & 16 \\ 0 & 0 & 0 & 0 & 1 & 14 \\ \end {array} }\)

$r_4 \to r_4 - r_5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccc{{|}}c} 1 & 1 & 0 & 0 & 0 & 52 \\ 0 & 1 & 1 & 0 & 0 & 43 \\ 0 & 0 & 1 & 0 & 0 & 18 \\ 0 & 0 & 0 & 1 & 0 & 16 \\ 0 & 0 & 0 & 0 & 1 & 14 \\ \end {array} }\)

$r_3 \to r_3 - r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccc{{|}}c} 1 & 1 & 0 & 0 & 0 & 52 \\ 0 & 1 & 0 & 0 & 0 & 25 \\ 0 & 0 & 1 & 0 & 0 & 18 \\ 0 & 0 & 0 & 1 & 0 & 16 \\ 0 & 0 & 0 & 0 & 1 & 14 \\ \end {array} }\)

$r_2 \to r_2 - r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {ccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & 27 \\ 0 & 1 & 0 & 0 & 0 & 25 \\ 0 & 0 & 1 & 0 & 0 & 18 \\ 0 & 0 & 0 & 1 & 0 & 16 \\ 0 & 0 & 0 & 0 & 1 & 14 \\ \end {array} }\)

$r_1 \to r_1 - r_2$

$\blacksquare$