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.


 

Dr Steve's Walking Speed

I try to take a walk every day. The walk is a loop, 2.2 miles long. I have "landmarks" at one mile and two miles. I am interested in how fast I walk, because I want the walk to be "aerobic" (exercises the heart and lungs). Usually aerobic speeds are given in miles per hour, but my stopwatch gives minutes and seconds; hence, conversion is needed.

To convert minutes to hours, divide by 60 (minutes per hour). For example, 15 minutes is a quarter of an hour, 30 minutes is half an hour.

For example, suppose I walk 1 mile in 18 minutes, 2 miles in 36 minutes, and 2.2 miles in 38 minutes.

Convert 18 minutes to hours: 18/60 = 0.3, exactly. So 1 mile in 18 minutes is 3.333 miles per hour. Similarly, 2 miles in 36 minutes is also 3.333 miles per hour. (Note: 333 is short for an endless string of "threes.")

Convert 38 minutes to hours: 38 minutes is 0.63333 hours. So, 2.2 miles in 38 minutes is 3.47 miles per hour (I probably speed up when the goal is in sight.)

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

My most recent walk was 1 mile in 19 minutes, 2 miles in 35 minutes and 50 seconds, and 2.2 miles in 38 minutes and 20 seconds. You need to convert the seconds to decimal minutes; 50 seconds is 0.8333 minutes, and 20 seconds is 0.3333 minutes

What was my walking speed in miles per hour for 1 mile, 2 miles, and 2.2 miles?

Answers in Rot13:

Ebhaqrq gb guerr qrpvzny cynprf: bar zvyr, guerr cbvag bar svir rvtug zcu, gjb zvyrf, guerr cbvag guerr sbhe avar zcu, 2.2 zvyrf guerr cbvag sbhe guerr guerr zcu.

.

Correction to previous post

If you don't know how many ounces are in a pound, adults can tell you.

This problem came up in real life. My wife Karen saw red bell peppers at Whole Foods for $5 per pound. Too expensive! She asked me to go to Super Value and buy some for less than $5 per pound. I went in and got two peppers, on sale, 2 for $3. When I got them home, I found that the two peppers, together, weighed 13 ounces. What was the price per pound? If you don't know how many ounces are in a pound, adults tell you.

Answer, in Rot13 code: Guerr qbyynef naq fvkgl avar pragf cre cbhaq.

A Day or So After the Summer Solstice

According to the US Naval Observatory, the Sun will rise on June 23 2014 at 05:27 Central Daylight Time (CDT) and will set at 21:04 CDT. From now on, days will be shorter. How long will the day be?

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

Answer: The times are given on a 24-hour clock. 21:04 is the same as 9:04 PM on a 12-hour clock. Note that the colon : is not a decimal point. Minutes, to the right of the colon, are base 60. For example, 5 minutes after 20:59 is 21:04. Hours, to the left of the colon, are base 12 or base 24. For example, 1 hour after 23:30 is 00:30.

We need to subtract 05:27 from 21:04. You can't take 27 from 04, so you "borrow" an hour from 21. When it crosses the colon, 1 hour becomes 60 minutes and 21:04 becomes 20:64. Now subtract 27 from 64 and get :37 and subtract 5 from 20 and get 15. The day is 15 hours and 37 minutes long.

Challenge: How long is the night?

Later in the month, we should expect the days to get shorter and nights to get longer.