Share

## Problems to Ponder

During his NCTM Presidency (2010-2012), Mike Shaughnessy selected an interesting math problem to present each month. These "Problems to Ponder" are great opportunities for teachers and their students to engage in reasoning and sense making.  Additional “ponderings” to spur your thinking were added with the solution in the following month's Summing Up. The following archive contains the original problems, additional ponderings, and a solution.

Archived "Problems to Ponder" and Solutions:

March 2012: It Takes Two to Geo-Tangle!
How many different polygons of area 2 can you construct on a 5 x 5 geoboard?

February 2012: Sleuthing the Neighborhood
A private detective is working a case and has managed to verify the following information...

January 2012: Under Lock and Key
In medieval times, the inhabitants of a remote village decided to lock the village valuables in a giant chest to protect them from marauding thieves.

How many zeroes occur at the end of the expanded numeral for 1000!? How about in general, for n!?

November 2011: Going Around in Circles
If we pick any two distinct points on a circle, and connect them with a chord, the chord will divide the interior of the circle into two distinct, nonintersecting regions...

October 2011: Water Bucket Conundrum
You are staying at a rural cabin, and the only method to get water is to draw it from a well. A 4-gallon bucket and a 9-gallon bucket are the only containers for carrying water to the cabin...

September 2011: Figuring out Figurate Numbers
The figure below shows the visual pattern for the first few triangular, square, pentagonal, and hexagonal numbers...

August 2011: Factor Craze
Here’s a Problem to Ponder that will help your students ease back into the groove at the start of the school year. Students usually know that prime numbers have exactly two factors...

July 2011: Looking Squarely at the Difference!
Here’s a “kick back and enjoy” summertime Problem to Ponder: Which whole numbers can be expressed as the difference of two perfect squares?

June 2011: It’s as Easy as Falling Off a Cliff, or Is It?
In a kingdom long ago, a king decided to let chance determine whether persons who committed major crimes...

May 2011: Circling Around
Suppose that the circles in the figure below are tangent to one another...

April 2011: It’s All About “Prod-Difs”—Vive la Différence and Multiply
Pick any four positive integers. For example, 5, 14, 17, and 23 give a collection of four positive integers...

March 2011: Interior Crossings
In the rectangular grids below, the diagonal touches the interiors of some of the squares in the grid...

February 2011: Tempted by Triangles
How many different triangles are in the figure below?...

January 2011: A Rematch for the Tortoise and the Hare
Some years after their famous race recounted in Aesop’s fable...

December 2010: The Good, the Bad, and the Ugly
In a re-creation of a famous 3-cornered duel...

November 2010: Ponder This Tangled Triangular Relationship!
In triangle ABC, segment AG from vertex A meets opposite side BC at a point G...

October 2010: A Howling Good Halloween Puzzle
How many times a week do you want to eat chocolate? (Pick more than one but less than 10 times)...

September 2010: Where Would You Sit in Your Neighborhood Café?
In a neighborhood café there are 10 seats in a row at the counter...

August 2010: Which Whole Numbers Can Be Written as the Sum of Consecutive Whole Numbers?
Consider, for example, that 6 = 1 + 2 + 3 and 15 = 4 + 5 + 6...

July 2010: Comparing Some Regions in a Square
In the figure below, quadrilateral ABCD is a square, and E is the midpoint of the side AD. How do the areas of regions I, II, III, and IV compare?...

June 2010: Does It Matter Which Winner You Saw?
Scenario: Students at your school have just finished competing in the qualifying round of a nationally sponsored contest on mathematical reasoning and sense making...