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

by : $134$

 * The Bag of Nuts

Solution
In the first to fifth bags in order, the number of nuts is:
 * $27, 25, 18, 16, 14$

Proof
Let $a$, $b$, $c$, $d$ and $e$ be the number of nuts in each of the first, second, third, fourth and fifth bags respectively.

We have:

We set up this system of linear simultaneous equations in matrix form as:


 * $\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}$

It remains to solve this matrix equation.

In reduced echelon form, this gives:

from which the contents of the bags can be read off directly.