Question No. 6 is the stumbling block;
No winners this month

Many of the entries in the January Math Puzzler competition got five of six answers correct. But, the answer to Question No. 6 eluded everyone.

The most common answer was 9 hours. And, it seemed to be the "logical" answer. But, Mr. Math Puzzler had a different idea of what "positioned in a straight line" meant.

He was able to get the satellites re-aligned along his straight line in half the time, or 4 1/2 hours.

Many of you thought of the "straight line" as a radius from the Earth to the outer edge of the orbits. That's the way the satellites showed in the illustration for the question.

But, Mr. Puzzler's "straight line" was a full diameter of the orbits, running on both sides of the earth. The illustration for the answer to Question No. 6 shows how that works.

Question No. 6 was the only one where all your answers were incorrect. For many of you, the other answers seemed to come easy - although Mr. Math Puzzler thought he was making the questions very challenging. .

For some tips by Mr. Math Puzzler on how to increase your chances of getting future answers right, see the February Math Puzzler story. (To get there quickly, just click here after you review the January answers that follow.)

Answers to January, 2001, Math Puzzlers

1. The clerk misunderstood the order for rope. He reversed the feet and inches and the customer got only 30% of what she ordered. What was the length of rope really ordered?

Answer: 9 ft, 2 inches

Explanation: There's a way to solve this with algebraic formulas. But, this is also one where starting with an "educated guess" can put you on the road to the right answer. One way to focus your guess is to assume the rope won't be higher than 11 feet. That's because the "reversed" number can't have an inch figure over 11.

So, start with 11 feet, 10 inches and work down. Eleven feet, 10 inches, when reversed comes out 10 feet, 11 inches. That doesn't fit the 30% comparison. By working this way, you'll eventually arrive at 9 ft, 2 in. A lot of you got there on this question.

 

2. Jerry went shopping. He spent half of what he had plus $5 for a shirt. Then, he spent half of what was left plus $5 for a mystery novel. Finally, he spent half of what was left plus $5 for stationery. Then, Jerry had nothing left. How much did he start with?

Answer: $70

Explanation: To get this correct answer takes a number of algebra steps. But, many of you found the correct answers. We start with X being the final answer.

 

3. There was a contest to guess how many beans were in a jar. Herb was off by 6, Sally was off by 5, James was off by 4 and Lucy was off by 3. Susie was correct. One guessed 19, one guessed 22, one guessed 29, one guessed 25 and one guessed 30. What was the right answer?

Answer: 25 beans

Explanation: This is another answer that can be found with "educated" trial and error. Arrange people by order, such as Herb, off 6; Sally, off 5; James, off 4, Lucy, off 3, and Susie, off 0. Then, starting with Herb, figure out which numbers off by 6. The only combination is 19 and 25. Then, check out Sally's off 5 number and you see 30 is 5 off of 25. Using 25 as the correct answer, James' off 4 number is 29 and Lucy's off 3 number is 22.

 

4. Jim climbed a rock wall at half a mile per hour and descended twice as fast. The trip took 12 hours. How far was it to the top?

Answer: 4 miles

Explanation: You can see that if going up takes twice as long as going down. That means two-thirds of total 12-hour round trip is (or 8 hours) is spent going up and one-third (or 4 hours) in coming down. You know you made progress at a half-mile per hour going up so the answer is 4 miles.

 

5. Fifteen knights were invited to a sumptuous meal at the castle. Before sitting down, each of the 15 knights shook hands with each of the other knights. How many hand shakes were needed?

Answer: 105

Explanation: The answer is arrived at by figuring a progression of handshakes. Knight No. 1 needs to shake hands with the other 14 knights. But, Knight No. 2 only requires 13 new handshakes because his shake with Knight No. 1 is already accounted for. Knight No. 3 only needs 12 new shakes, because his handshakes with Knights No. 1 and No. 2 are accounted for in those knights' shake totals. This continues until Knight No. 15 has no new handshakes because his shaking is accounted in the other 14 totals. And 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1 new handshakes adds up to 105.

 

6. Different things orbit the earth at different speeds and distances. For example, satellites and space instruments released by the space shuttle are only several hundred miles away from earth, while communication satellites circle at a distance of about 22,300 miles.

In this puzzle, Satellite X-1 orbits our planet once every nine hours, Satellite Beta once every 4 1/2 hours and Satellite Parking once every three hours. At time zero, the satellites are positioned in a straight line. How long will it take for all three objects to position themselves again in a straight line?

Illustration A

Answer: 4 1/2 hours

Explanation: Compare the original question illustration (A) with the answer illustration (B).

Illustration B

If you ran the satellites around until they lined up like Illustration A, the lapsed time would be 9 hours. But, if you consider alignment along the whole diameter it would take only 4 1/2 hours. That's because Satellite X-1 only needs to make a half-orbit to get aligned to Satellite Beta, which makes a whole orbit in that time. Then, Satellite Parking aligns at the bottom of the orbit (with Parking) by going around 1 1/2 times in its 4 1/2 hours.