## 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

## Distinct Triangles

How many distinct triangles can be constructed by choosing three vertices from among the corners of a unit cube? Source: NCTM Mathematics Teacher, February 2008 Solution We construct four triangles from three corners of a face for a total of … Continue reading

## Measure of Angle Z

Find the measure of the angle in the diagram here, given that the two horizontal segments are parallel. Source: NCTM Mathematics Teacher, February 2008 Solution Complete a seven-sided concave polygon by drawing a perpendicular line to the two parallel lines … Continue reading

## Two Even Digits

The number is a four-digit number having two zeros and two even digits that are not zeros. How many four-digit numbers besides can also be described this way? Source: NCTM Mathematics Teacher, February 2008 Solution First, choose a way to … Continue reading

## Product of Factorials

Solve for if . Source: NCTM Mathematics Teacher, February 2008 Solution Answer:

## Equilateral Triangle and Regular Hexagon

What is the ratio of the area of an equilateral triangle to that of a regular hexagon inscribed in the same circle? Source: NCTM Mathematics Teacher, February 2008 Solution The hexagon is made up of six congruent isosceles triangles three … Continue reading

## Three Consecutive Odd Numbers

Find three consecutive odd numbers such that the sum of their squares is a four-digit number with all four digits the same. Source: NCTM Mathematics Teacher, February 2008 Solution The squares of odd numbers are odd and the sum of … Continue reading

## Divide Evenly

Consider the set of positive integers for which there exists an integer such that evenly divides both and . Find the sum of the elements of . Source: NCTM Mathematics Teacher, August 2006 Solution The integers that are divided evenly … Continue reading