Counters On A Chess Board

Logic Level 2

What is the maximum number of counters you can place on an $8 \times 8$ chessboard given that each row, column, and the two main diagonals contain 5 or fewer counters?

The answer is 40.

2 solutions

Sam Bealing
Jul 10, 2016

As we have $8$ rows and $5$ or fewer counters per row we have an upper bound of $5 \times 8=40$ so all we need to do is show this can be achieved:

$\color{#3D99F6}{\text{X}}$				$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$
$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$			$\color{#3D99F6}{\text{X}}$		$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$
$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$		$\color{#3D99F6}{\text{X}}$		$\color{#3D99F6}{\text{X}}$		$\color{#3D99F6}{\text{X}}$
$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$		$\color{#3D99F6}{\text{X}}$
$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$				$\color{#3D99F6}{\text{X}}$
	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$		$\color{#3D99F6}{\text{X}}$
		$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$
		$\color{#3D99F6}{\text{X}}$		$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$	$\color{#3D99F6}{\text{X}}$

Moderator note:

What is the motivation that lead to this construction?

Yes , in order to prove it you can just show that the upper bound can be achieved but what is done by this is jsut showing and not explaining what makes such constructions work.

Nonetheless , if you try to fully understand it can you say what makes it work and how you arrived at such a construction or , if there is some rigorous way to construct the possible solutions because by this kind of understanding you will get truly into the depths and principles of such configurations anyway.

A A - 4 years, 11 months ago

No. Of rows * maximum no. Of pawns. In one row =5*8=40

OS overspike - 3 years, 1 month ago

Don Weingarten
Feb 5, 2019

The maximum of 5 in each of 8 rows implies an upper limit of 40. Experimentation shows various ways of placing 40 counters to satisfy the given conditions.

Counters On A Chess Board

The answer is 40.

2 solutions

Moderator note:

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