Retired high school math teacher.

## Difference of Sequences

For the sequence , what is the difference between the mean of the sequenceâ€™s first terms and the mean of its first terms? Source: NCTM Mathematics Teacher February 2008 Solution If we add the first terms by pairs we end … Continue reading

## Sums of Reciprocals

The sum of the positive divisors of is . Find the sum of the reciprocals of the positive divisors of . Source: NCTM Mathematics Teacher, February 2008 Solution Given that , the number of divisors is . The first twelve … Continue reading

## Divisible by Units Digit

The number is divisible by its units digit . How many whole numbers less than are divisible by their respective units digit? Source: NCTM Mathematics Teacher, February 2008 Solution Nine numbers: divisible by themselves Six numbers: divisible by Six numbers: … Continue reading

## Sum of 7’s

In the sum of the expression , what digit will be in the tens place? Source: NCTM Mathematics Teacher, February 2008 Solution We are adding nineteen numbers from the smallest to the largest . Note that there are nineteen in … Continue reading

## Numerical Puzzle

The five-square numerical puzzle below requires the vertical column to be filled with the digits from a three-digit integral power of and the horizontal row to be filled with the digits from a three-digit integral power of . What digit … Continue reading

## 4-Digit Numbers Divisible by 7

Given the digits , and , find the probability that a four-digit number formed by using each of them only once is divisible by . Source: NCTM Mathematics Teacher, February 2008 Solution There are possible four-digit numbers using each only … Continue reading

## Two-mile Long Train

How long will it take a two-mile long train traveling miles per hour to travel completely through a mile-long tunnel? Source: NCTM Mathematics Teacher, February 2008 Solution The diagram shows that for the train to clear the tunnel, point must … Continue reading

## Triangular Array of Numbers

If you continue the triangular array of numbers shown in the figure, what number would be directly below ? Source: NCTM Mathematics Teacher, February 2008 Solution Note that the leading numbers increase by , etc. If we follow this pattern, … Continue reading

## Smaller Factor

Given a three-digit number, interchange the units digit and the hundreds digit. The product of the original number and the new number is . What is the smaller three-digit factor? Source: NCTM Mathematics Teacher, February 2008 Solution Since the product … Continue reading

## Non-Congruent Triangles

Among the triangles of the previous problem (repeated below), how many distinct (non-congruent) triangles exist? Previous Problem: How many distinct triangles can be constructed by choosing three vertices from among the corners of a unit cube? Source: NCTM Mathematics Teacher, … Continue reading