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The original puzzle
Some restaurants make clients pay according to their means. There is a dollar entrance fee, the meal costs half of the money
you have, and there is a tip of a dollar before leaving. How much money did you start with if you have none left after visiting
4 restaurants ?
The Solution
This enigma is quite similar to that of the apples and, once again, we start at the end -
1. At the fourth restaurant you paid "$1 entrance, 1/2 of your money, and $1 tip"
and had no money left.
"1/2 of your money" was evidently $1 to leave the $1 tip
so you had $2 before paying the meal plus $1 to get in,
$3 on arriving at the restaurant.
2. At the third restaurant you paid "$1 entrance, 1/2 of your money, and $1 tip"
and had $3 left.
"1/2 of your money" was evidently $4 to leave the $1 tip
so you had $8 before paying the meal plus $1 to get in,
$9 on arriving at the restaurant.
3. At the second restaurant you paid "$1 entrance, 1/2 of your money, and $1 tip"
and had $9 left.
"1/2 of your money" was evidently $10 to leave the $1 tip
so you had $20 before paying the meal plus $1 to get in,
$21 on arriving at the restaurant.
4. At the first restaurant you paid "$1 entrance, 1/2 of your money, and $1 tip"
and had $21 left.
"1/2 of your money" was evidently $22 to leave the $1 tip
so you had $44 before paying the meal plus $1 to get in,
$45 on arriving at the restaurant.
Mathematics
The formula for the money required, in terms of the number of reataurants,
is not as simple as it was for the apples,
the $1 entrance charge and the $1 tip tripling the rate of increase.
If "n" represents the number of restaurants,
the number of dollars at the very start will be 3(2n - 1).
Once again the usual 2n - 1 formula.
For 10 restaurants you would need $3069, a tidy sum !
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