Algebraic Solution to "Ask Marilyn" Apple Problem of April 21, 2019
"Ask Marilyn" of Parade Magazine posed the following problem:
A prince asked a king if he could marry his daughter. The king gave him three baskets. “Pick enough apples so you can put half of them plus ½ apple more in the first basket. Then put half of the remaining apples plus ½ apple more in the second one. Then put half of the apples left plus ½ apple more in the third basket. If you have one apple left for my daughter, I will judge you smart enough to marry her.” How many apples should the prince pick?
She gave the answer as 15 which is correct, but didn't provide an algebraic method of the solution. Here it is.
Assume x is the total number of apples.
Basket #1 will contain half of the total # of apples plus half an apple:
Basket #2 will contain half of the remaining apples plus half an apple:
Basket #3 will contain half of the remaining apples plus half an apple:
There is one apple left for the princess, so the final equation is:
Basket #1 gets 8, #2 gets 4, #3 gets 2, and the princess gets the 1 remaining apple. It also shows that 15 is a unique solution to the puzzle.
JLF
Derived a simple generic equation below. Seems to work. The term 2^n represents 2 to the power of n.
ReplyDelete(x+1-2^n)=r * 2^n
Where
x is number of apples to pick from basket
n is number of baskets
r is number of apples remaining
Example:
n=3, r=1 then you get x as 15
n=3, r=2 then you get x as 23
n=4, r=1 then you get x as 31