Bargain Price For a Dress

My wife is very good at getting bargains on clothes. She showed me a new dress, it was very nice, and showed me the price tag, $128. She asked me to guess what she payed for it. It told her I could guess the price in seven guesses or less.

Me: "was it more or less than $64?" Guess #1
My wife: "less"
Me: "was it more or less than $32?" Guess #2
My wife: "less"
Me: "was it more or less than $16?" Guess # 3
My wife: "less"
Me: "was it more or less than $8?" Guess #4
My wife: "more"
Me (to myself) "OK, I know it cost between $8 and $16. What is the midway point between $8 and $16? OK, it is $12."
Me: "was it more or less than $12?" Guess #5
My wife: "more"
Me (to myself, again) "OK, I know it cost between $12 and $16. What is the midway point between $12 and $16? OK, it is $14."
Me: "was it more or less than $14?" Guess #6
My wife: "less"
Me: "You paid $13 for it. Guess #7
My wife: "Correct."

My wife had payed about one tenth of the original price and had bought a very nice dress, suitable for wearing to the theater, for example.

----------------------------

Honestly, dear reader, I am not making this up. Fortunately, 1) the original price was $128 and 2) I have memorized a great many powers of 2 and knew that 128 is 2 to the seventh power. Hence, by what is called a binary search, I knew I could guess the sale price in seven guesses.

I first learned about this in 1955 when I was a guest at a party where the mathematical genius Norbert Weiner was the guest of honor. He was talking about the game "20 Questions" and said one could narrow a million possibilities down to one answer by using 20 well posed questions, because 2 to the 20th power is slightly larger than a million; in fact it is
1 048 576.

Minneapolis Public Schools has an annual 5th grade Math Contest. Some of the problems in these contests can be solved only by guessing and checking. I hope in future posts to give some examples which can be solved by using a table with only six or seven rows.