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(2^{n}  1). Once again the usual 2^{n}  1 formula. For 10 restaurants you would need $3069, a tidy sum !
