The number of entries was up in the Math Puzzler competition in October. But, for the first time in awhile, there were no entrants who got all the answers correct.

That's the first time in several months that Mr. Math Puzzler has stumped all the entrants.

For awhile, we thought Mr. Puzzler had gone soft on his questions. But, in October, he put in some Puzzlers that required educated guessing and some ingenuity, rather than just established formulas.

In October, we also started again to get group entries from individual schools.

That must mean that some of you have been asking your math teachers to give some extra credit if you enter the Math Puzzler competition.

Wayne Hesse is our Mr. Math Puzzler. He's an eighth grade teacher at Green Park Lutheran School in south St. Louis County.

For years, he's given his math students extra "fun" questions to finish after their regular assignments are done. In class, those puzzles might be actual wooden or metal puzzles or they might be paragraph problems on paper.

His students can do those extra puzzles for both fun and extra credit.

That's what the Young Saint Louis.com Math Puzzlers are all about. Learning math while having fun. And without the pressure of getting a formal grade.

Be sure to ask your math teachers if he or she will give extra credit for entry in the YSL.com competition. We have new questions every month.

Each month, we remind new entrants to check out some previous questions and answers to learn how Mr. Math Puzzler thinks. The Puzzlers started over two years ago so you have plenty of examples of both the questions and the answers.

Just click on to the Past Stories tab at the top of the homepage. Pick any month after September, 2001, and you'll have examples of questions and answers. When you are ready for the November, 2003, questions, click here.

Remember, if you get all six answers correct, we publish your name next month along with the November answers.

Also, as an added incentive, we put all entries with six right answers into a hat. Then, we draw up to three and award $10 Border's book certificates to them.

The October Math Puzzlers answers

1. Two towns are linked by a railroad. Every hour on the hour a train leaves each town for the other town. The trains all go at the same speed and every trip from one town to the other takes five hours. How many trains are met by one train during a one-way trip?

Answer: 11 trains

The explanation: Since the trains are traveling toward each other, one train will meet another every 30 minutes. There are nine 30-minute points in a five-hour period. Then, there will be one other train in the depot when our train leaves and another in the depot at the destination point. That adds to 11 trains met.

 

2. Timothy spent all his money in five stores. In each store, he spent $1 more than half of what he had when he came in. How much did Timothy have when he entered the first store?

Answer: $62

The explanation: The best way to do this is to start at the end of the sales spree. To be able to have something to spend in the fifth store, Timothy would need to have had $3. (That's $2 being half of the amount from the previous store plus the $1 extra.) Then, we go backwards. He'd have entered the fourth store with $6 plus the $1 extra or a total of $7. Using this same pattern, he'd have had $14 plus $1 or $15 in the third store; then $30 plus $1 of $31. Double the $31 and he started with $62.

 

3. How many ways can you read POP off the diagram below? Letters must touch each other horizontally, vertically or diagonally. Any P can be both the first and letter of a single POP? (Hint: Remember, you can spell backwards as well as use some back-and-forth spelling.)

              P
            P O P
          P O P O P
            P O P
              P

Answer: 64

The explanation: There is no formula for this. But, starting with the P's on the four peaks of the figure, you can achieve four POPs each for a total of 16. Then, you can find 8 POPs with the four mid-line P's or 32. Then, for the center P, you can find 16 other POPs. That's a total of 64.

 

4. Which three digits are represented by X, Y and Z in this sum?

               XXXX
               YYYY
               ZZZZ
              -----
              YXXXZ

Answer: X=9, Y=1, Z=8

The explanation:

             9 9 9 9
             1 1 1 1
             8 8 8 8
            --------
            1 9 9 9 8

 

5. Consider all the whole numbers from zero through one billion. What is the sum of all the digits needed to write down these numbers?

Answer: 40,500,000,001

The explanation: This is a question that uses a very old formula. The way to find the number of digits in all those numbers is to add the digits in a series of lines that starts with the first digit and the digits in the last number before one billion, which is 999,999,999.

1. That looks like this:
   0 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = 81
2. The second pattern is:
   1 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 8 = 81
3. Continuing, you find there are 500,000,000 million pairings, all totaling 81 digits.
4. Multiply 500,000,000 by 81 and you get 40,500,000,000.
5. But, you need to add one extra digit to represent the final digit in one billion, thus the answer of 40,500,000,001.

 

6. In a stable, there are men and horses. In all, there are 22 heads and 72 feet. How many men and how many horses are in the stable?

Answer: 8 men, 14 horses

The explanation: You can set up two formulas to represent the two parts to this question. We'll use M for men and H for horses.

The "head" quotation: H x M = 22

The "feet" quotation: 4H + 2M = 72

Then, multiply the "head" quotation by -2 so we can eliminate one portion:
-2 (H + M) = -2 (22) becomes -2H - 2M = -44

Then subtract the "feet" quotation:

     -2H - 2M = -44
      4H + 2M =  72
      -------   ---
      2H      =  28
      --         --
       2          2

            H = 14

If there are 14 horses, there have to be 8 men.