Henry Ernest Dudeney/Modern Puzzles/129 - The Circle and Discs/Solution
Modern Puzzles by Henry Ernest Dudeney: $129$
- The Circle and Discs
- During a recent visit to a fair we saw a man with a table,
- The circular discs were all of the same size, and each, of course, smaller than the red circle.
- he showed that it was "quite easy when you know how," by covering up the circle himself without any apparent difficulty,
- but many tried over and over again and failed every time.
- It was a condition that when once you had placed any disc you were not allowed to shift it,
- otherwise, by sliding them about after they had been placed, it might be tolerably easy to do.
- Let us assume that the red circle is six units in diameter.
- Now, what is the smallest possible diameter for the five discs in order to make a solution possible?
Solution
The dotted lines represent the red circle and a regular pentagon inscribed within it.
The center of this circle is the point $C$.
Find the point $D$ which is equidistant from $A$, $B$ and $C$.
With radius $AD$, draw the circle $ABC$.
Similarly construct the points $E$, $F$, $G$ and $H$.
These points are the centers of the circles representing the discs with which the red circle is covered.
We are given that the red circle is $6$ units in diameter.
The diameter of the discs is then just under $4$ units in diameter.
Covering is possible if the ratio of the two diameters is greater than approximately $0.6094185$.
In the case above, where all five discs reach the center of the red circle, the ratio is the golden mean $\approx 0.6180340$.
Proof
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Sources
- 1915: Eric H. Neville: Solutions of Numerical Functional Equations, illustrated by an account of a Popular Puzzle and its Solution (Proc. London Math. Soc. Ser. II Vol. 14)
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Geometrical Problems: Various Geometrical Puzzles: $129$. -- The Circle and Discs
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Geometrical Problems: Circle Puzzles: $287$. The Circle and Discs