Closing the Teach For America Blogging Gap
May 01 2012

More Problem Solving (NCTM Takeaway 3)

By popular demand (and because I need somewhere to organize this for myself), I’m going to put up a series of posts about what I learned at the NCTM conference. If you’re a math teacher, I hope this is as valuable for you as it is for me!

(If you’re not a math teacher, I promise I’ll include “NCTM Takeaway” in all the titles so you’re warned in advance. These are posts you definitely don’t have to be reading. I also promise I’ll return to regular posts soon enough.)


I left all my notes at school, so today you get more problem solving puzzles, from a Logic session by Diana Cheng. I carry these around with me because I’m actually really enjoying working through them myself. The first three would work well for middle school kids, but the “Numerical Pattern Puzzles 2″ are much more challenging.

I think I’ve figured out #1-6, but #4 took me so long that I’m only now getting to the last two problems. If you get an answer to anything in the Puzzles 2 section, I’d love to see it so I can check my own work. (Or if you were really quick on #4, I’d love to hear how you did that too!) You could leave it in the comments, or save people the spoiler and email me at [email protected]


Numerical Puzzles 1

1) A printer uses 207 digits to number the pages of a book. How many pages are numbered in the book?


2) In a video game, capturing a large jewel gives you 7 points. Capturing a small jewel is worth 3 points. What is the largest score that is impossible?


3) A jar is full of jellybeans If you count them by either 2s, 3s, or 5s, there is always one jellybean left over. If you count the jellybeans by 7s, there will be none left. How many jellybeans are there in the jar?

Numerical Puzzles 2

4) A man goes to the bank and cashes a check. The teller read the numbers on his check wrong, and gives the man cents for dollars and dollars for cents. The man then sees a friend of his who owes him $10.00. After his friend pays him back, the man buys a soda for 66 cents. The man counts the money he has now and discovers that he has twice as much money as the face value of the original check. What was the face value of the original check.


5) A large purse is full of coins. If you count them by 13s, 23s, or 31s, there will be one left over. If you count them by 73s there will be none left over. How many coins are there in the purse?


6) License plates are issued sequentially; in a state where license plates consist of three letters followed by three numbers, the following sequence could be issued: ABC998, ABC999, ABD000, ABD001,… The last license plate issued in this state was AZY987. No letters or numbers in this license plate are repeated. How many license plates must be issued after this one before another plate will also have no repeated letters or numbers?


7) One morning, the parson tells his sexton: “I met three people on the way here today. The product of their ages is equal to 2450. The sum of their ages is the double of yours. What are their ages?” In the afternoon the sexton told the parson that he could not answer the question as asked. The parson added: “I can only say that one of the three is older than me.” How old is the parson?


8 ) In the country of Nacirema, there are only two types of coins, worth 13 cents and 17 cents. Can you buy a 20 cent newspaper in Nacirema and receive exact change? How?


Seriously – if you do these problems (especially 4-8) it would be a big favor to me if you’d share solutions so I can check mine. Thank you :-)





3 Responses

  1. Gary Rubinstein

    For #8, the easiest way is to notice that 20 is a multiple of 4 and 17-13=4 multiply both sides by 5 to get 17*5-13*5=85-65=20

    For a number that is not a multiple of 4, something like 23, it would be more difficult, but still possible. Anytime two numbers are relatively prime, like 13 and 17, it is possible to find integers such that, in this case, 13x+17y=1 using something called the extended euclidean algorithm. By ‘inspection’ since 13*4=52 and 17*3=51, 13*4-17*3=1. Then if you multiply both sides by 20, you get 13*80-17*60=20 so you can pay 13*80, which is 1040 and get back 17*60=1020 so you’ve paid 20 altogether. This process would work for any amount.

  2. Lucas

    4. Suppose the check has a value of x dollars and y cents. The teller would then give the man y dollars and x cents. Notice that both x and y must be non-negative integers, and y must be less than or equal to 99. The dollar amount of the check is x + y/100. The dollar amount of money the man has at the end is y + x/100 + 10.00 – 0.66 = y + x/100 + 9.34. The amount of money he has at the end is twice the value of the check, so y + 0.01x + 9.34 = 2 (x + 0.01y), and y = (1.99x – 9.34)/0.98. Calculate y for each non-negative integer value of x such that y is greater than zero and less than or equal to 99. The only solution such that y is also an integer is x = 50, y = 92, so the value of the check is $50.92 and the teller gives the man $92.50. His friend gives him $10.00, so he has $102.50, and then he buys a soda for $0.66, so he has $101.84, which is indeed twice $50.92.

    • Lucas

      * …such that y is greater than or equal to zero…

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