You have been given the job of designing the "roll a penny" game for your school's summer fair.

Background: The player is given a small 'shoot' down which they roll a
coin on to a large board marked with a square grid of lines. The coin
rolls across the board and when it stops, and falls flat, it will lie
either, entirely within a square - not touching a line, or only partly
within a square and across a line. If it is entirely within a square the
player wins. If it is across a line the player loses.

a) Your first effort should provide the player with an equal probability
of winning and losing. If the coin has a radius of one unit, what is the
side length of the square that gives an equal win / lose probability? You
can ignore any element of player skill and any complications associated
with line thickness etc.

b) The school vicar, who knows more about such things than he likes to
admit, points out that the game would be more interesting if the prize
could be made larger. The headmaster, responsible for school finances,
wants the game to at least break even. You are asked to modify your design
such that the player only has a 1 in 10 chance of winning on each roll. If
the coin has a radius of one unit, what is the side length of the square
that gives an 1 in 10 win, 1 in 9 lose, probability? You can still ignore
all complications.

I remember playing the game at primary school in the 1960s and working out
the probability at secondary school, when introduced to the subject, some
fifty odd years ago.

--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Wed, 4 Jun 2025 08:53:15 -0000 (UTC), David Entwistle wrote:

that gives an 1 in 10 win, 1 in 9 lose, probability? You can still
ignore all complications.

Sorry...

1 in 10 win, 9 in 10 lose probability.


--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 6/4/2025 1:53 AM, David Entwistle wrote:

You have been given the job of designing the "roll a penny" game for your school's summer fair.

Background: The player is given a small 'shoot' down which they roll a
coin on to a large board marked with a square grid of lines. The coin
rolls across the board and when it stops, and falls flat, it will lie
either, entirely within a square - not touching a line, or only partly
within a square and across a line. If it is entirely within a square the player wins. If it is across a line the player loses.

a) Your first effort should provide the player with an equal probability
of winning and losing. If the coin has a radius of one unit, what is the
side length of the square that gives an equal win / lose probability? You
can ignore any element of player skill and any complications associated
with line thickness etc.

b) The school vicar, who knows more about such things than he likes to
admit, points out that the game would be more interesting if the prize
could be made larger. The headmaster, responsible for school finances,
wants the game to at least break even. You are asked to modify your design such that the player only has a 1 in 10 chance of winning on each roll. If the coin has a radius of one unit, what is the side length of the square
that gives an 1 in 10 win, 9 in 10 lose, probability? You can still ignore all complications.

I remember playing the game at primary school in the 1960s and working out the probability at secondary school, when introduced to the subject, some fifty odd years ago.

Spoiler?
Spoiler?
Spoiler?
Spoiler?
Spoiler?
Spoiler?
Spoiler?
Spoiler?
Spoiler?
Spoiler?
Spoiler?
Spoiler?
Spoiler?

When the game is won, the center of the penny will fall on a square
region within each grid square. If the side of the grid square is a, the
side of the inner square is a-2, since a penny has a radius of 1. The
ratio of this inner square's area to the grid square is the probability
of winning ("P"), (a-2)^2 = P a^2. This results in the quadratic equation: (1-P)a^2 - 4a + 4 = 0
Solving for a,
a = (2 + 2 sqrt(P))/(1-P)

For a probability of 0.5 (win half the time), a = 6.83 (approx.)

For a probability of 0.1 (win 1/10 of the time), a = 2.92 (approx.)

--
Carl G.


--
This email has been checked for viruses by AVG antivirus software.
www.avg.com

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Wed, 4 Jun 2025 11:52:08 -0700, Carl G. wrote:

Spoiler?

Well done. It took me a few goes, but I eventually got the right answer
again, many years after first working it out.

--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Thu, 05 Jun 2025 13:02:51 -0400, Charlie Roberts wrote:

You can avoid the quadratic. Given that (with r as the penny radius)

Nice. I hadn't seen that myself.


--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Wed, 4 Jun 2025 08:53:15 -0000 (UTC), David Entwistle wrote:

You have been given the job of designing the "roll a penny" game for
your school's summer fair.

There is a possibly(?) related question "195. - Lady Belinda's Garden" in
Henry Dudeney's Amusements in Mathematics, for which I haven't yet arrived
at a solution.

... "One of her gardens is oblong in shape , enclosed by a high holly
hedge, and she is turning it into a rosary for the cultivation of some of
her choicest roses. She wants to devote exactly half of the area of the
garden to the flowers, in one large bed, and the other half to be a path
going all round it of equal breadth throughout"...

There is then a picture showing one rectangle inside another rectangle,
with a constant-width-space, representing the path, between the two.

"How is she to mark out the garden under these simple conditions" She has
only a tape, the length of the garden, to do it with, and, as the holly
hedge is so thick and dense, she must make all her measurements inside.
Lady Belinda did not know the exact dimensions of her garden, and, as it
was not necessary for her to know, I also give no dimensions. It is quite
a simple task no matter what the size or proportions of the garden may be.
Yet haw many lady gardeners would know just how to proceed? The tape may
be quite plain - that is, it need not be a graduated measure".

--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Fri, 6 Jun 2025 13:18:41 -0000 (UTC), David Entwistle wrote:

There is a possibly(?) related question "195. - Lady Belinda's Garden"
in Henry Dudeney's Amusements in Mathematics, for which I haven't yet
arrived at a solution.

There is a partial on-line version of this question including the image,
but without attribution here:

https://www.pedagonet.com/mathgenius/test195.html

--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Sat, 07 Jun 2025 10:28:40 -0400, Charlie Roberts wrote:

I think the trickiest part, for me, was selecting the root to keep. I >>missed that at the initial go and spent a lot of time mulling over the >>positive root till it hit me.

Well done.

Dudeney has Lady Belinda marking out a quarter sized square and then
performing measurements on that, but the result is just the same.

--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)