# "Hare and Tortoise" Energy Mechanic

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DarkDream
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Joined: 12/31/1969

I recently purchased David Parlett's "The Oxford History of Board Games." I did not realize that David created the "Hare & Tortoise" board game that has sold millions over the years.

In chapter 6, he explains that the

Quote:
cost of movement [in terms of carrots] increases geometrically with the number of spaces moved. To move

2 forward costs 3 units (1+2)
3 forward costs 6 units (1+2+3)
4 forward costst 10 units (1+2+3+4)

. . . and so on.

The beauty of this formula is that there is a total of 63 squares on the board and 65 units of energy to spend. He goes on to say:

Quote:
From the Start, therefore you can get Home in 64 moves by the expenditure of 1 unit at a time. On the other hand, you could get nearly a quarter of the way round on the first two turns, leaving you well in advance but utterly played out, have spent either all 65 units on moves of ten or four, or 64 moves of seven and eight.

What this amounts to is the player having to find a balance of going to fast or going to slow:

Quote:
It was at this point in the originally abstract development of the game that the title of Hare & Tortoise naturally suggested itself, since skill consists in not haring ahead so fast as to run out of energy too soon, nor lagging tortoise-like behind so as to be unable to catch up when necessary.

I was thinking of using this geometric formula of moving n forward: (n^2 + n)/2. However, instead of having a large amount of energy units (like 65), I was thinking of having a smaller amount where you would have the probability of loosing an energy unit dependant on the speed you were traveling at by using this geometric formula. For example, if you travel at speed 1 (the max is speed 10 for a total amount of 55 in the equation above) the probability of loosing an energy unit is 1/55. If traveling at speed 2, the probability of loosing an energy unit is 3/55 and so on.

Does anyone have an idea to reflect a geometric distribution of probability using dice? If I had a 55 sided die I could do it, but I don't they exist out there. Anyone have any ideas besides creating 55 cards with different distributions of 1-10 on it?

Thanks,

--DarkDream

Joe_Huber
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Joined: 12/31/1969
Re: "Hare and Tortoise" Energy Mechanic

DarkDream wrote:
Does anyone have an idea to reflect a geometric distribution of probability using dice? If I had a 55 sided die I could do it, but I don't they exist out there. Anyone have any ideas besides creating 55 cards with different distributions of 1-10 on it?

How about approximating with a 20 sided die?

Speed 1 - never fail.
Speed 2 - fail only on a 1.
Speed 3 - fail on a 1 or 2.
Speed 4 - fail on a 1-4.
Speed 5 - fail on a 1-6.
Speed 6 - fail on a 1-8.
Speed 7 - fail on a 1-10.
Speed 8 - fail on a 1-13.
Speed 9 - fail on a 1-16.
Speed 10 - always fail. (Well, OK, I assume you want _some_ chance of success; this is just the logical conclusion of what you've laid out...)

Joe

Anonymous
"Hare and Tortoise" Energy Mechanic

If you don't care to be 100% accurate, you could use percentile dice (2d10) and come reasonably close:

1 = 2%
2 = 5%
3 = 11%
4 = 18%
5 = 27%
6 = 38%
7 = 51%
8 = 65%
9 = 82%
10 = 100%

GeminiWeb
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Joined: 07/31/2008
"Hare and Tortoise" Energy Mechanic

Good to see others were taking a closer look at what you were asking ... I saw

Quote:
Does anyone have an idea to reflect a geometric distribution of probability using dice?

and then decided we were talking about the Geometric probability distribution, so off I went ... the Geometric probability distribution describes the probability of getting so many failures before your first success (assuming independent trials):

Probability of x failures = p * (1-p)^x

where p = probability of success

[ mean = (1-p)/p ]

I wasn't quite sure how this fitted ... so I went back here and realised I hadn't interpreted the question in the first place . This however would be useful for working out things like average number of turns you could stay at a speed without losing a unit ...

Oh well, I suppose thats the price I pay for having a degree in statistics and not properly reading the question ...

DarkDream
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Joined: 12/31/1969
"Hare and Tortoise" Energy Mechanic

Joe,

I went ahead with a d20 scheme on my own and came up with this:

1 - never
2 - 1
3 - 1-2
4 - 1-4
5 - 1-5 (This differs from yours with a probability of 25% opposed to 30%. 21/55 is 27.27% -- I took the closer value).
6 - 1-8
7 - 1-10
8 - 1-13
9 - 1-16
10 - Always

I think I will use 1-6 for speed 5, like you did as it smoothes out the distribution more (5 looks odd).

Thanks for the input.

--DarkDream

Joe_Huber
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Joined: 12/31/1969
"Hare and Tortoise" Energy Mechanic

DarkDream wrote:

5 - 1-5 (This differs from yours with a probability of 25% opposed to 30%. 21/55 is 27.27% -- I took the closer value).
...
I think I will use 1-6 for speed 5, like you did as it smoothes out the distribution more (5 looks odd).

That's exactly the conclusion I came to, and why I used 1-6 - instead of going up by 1-2-3-4-5-6-etc. (as originally) it's 0-1-1-2-2-2-3-3-4, rather than the clunky 0-1-1-2-1-3-2-3-3-4.

Joe