Reshuffling knights - castle defenders

You can position 10 defenders of a square castle so that on every side there will be 5 men.

Yes, indeed, and here is a solution:

 5 0 0 0 0 0 0 0 5

This is an extreme situation in the sense that, if one of the knights falls in the battle, there is no one to take his place. So that with every casualty, the number of defenders on a side is decreasing. Let's consider a different variant:

 3 5 3 5 0 5 3 5 3

where 32 knights defend a king (denoted by 0 in the middle square) in such a manner as to have 11 fighters on every side. Now, after a ferocious siege, 4 knights were forced off the wall. Is it possible to position the remaining ones such as to still have 11 fellows on a side? Actually, the king, alarmed by the enemy's persistence, decided to go a step further and rearrange the remaining defenders so as to have 13 knights on the side. However, one knight among the wounded just before heaving his last breath and leaving the king's service for ever, suggested an arrangement of 14 knights on the side. What is it?

Remark

Once we noticed that, for a given number of knights, various arrangements are possible, other questions may be asked quite naturally. Is 14 on a side is the maximum number possible for a group of 28? What is the minimum? How many (or how few) are needed to have 14 on each side? These questions set the old problem into the framework of Linear Programming.

References

The puzzle is actually very very old. The last I heard of it it was as told by Scheherazade on the one thousand and fifth night in a new book by R. Smullyan.

References

1. Ya. I. Perelman, Fun with Maths and Physics, Mir Publishers, Moscow, 1988
2. R. Smullyan, The Riddle of Scheherazade and Other Amazing Puzzles, Ancient & Modern, Alfred A. Knopf, 1997
3. D. Wells, The Penguin Book of Curious and Interesting Puzzles, Penguin Books, 1992

A Sample of Optimization Problems

• Mathematicians Like to Optimize
• Isoperimetric Theorem and Inequality
• Viewing a Statue: the Problem of Regiomontanus
• Fagnano's Problem
• Minimax Principle Demonstration
• Maximum Perimeter Property of the Incircle
• Extremal Problem in a Circular Segment
• Optimization in Four Variables with Two Constraints
• Daniel Dan's Optimization in Three Variables
• Problem in a Special Trapezoid
• Cubic Optimization with Linear Constraints
• Cubic Optimization with Partly Linear Constraints
• Problem M317 from Crux Mathematicorum
• Find the Maximum and Minimum of a Function
• Area of Isosceles Triangle
• Minimum of Cotangents from Saint Petersburg
• Copyright © 1996-2018 Alexander Bogomolny

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