# Counting

## Fibonacci Word Problems I: Basic

Here and next week, we’ll look at a collection of word problems we have seen that involve the Fibonacci sequence or its relatives, sometimes on the surface, other times only deep down. The first set (here) are direct representations of Fibonacci, while the second set will be considerably deeper. A 20-foot walkway The first, from …

## Cutting Up Space Using n Planes

As the capstone of this series on counting, lets look at something that’s a little harder to count by drawing: What is the maximum number of regions into which all of 3-dimensional space can be divided by n planes? We’ll look at two significantly different perspectives. Working up through the dimensions The first question is …

## Cutting Up a Circle II: Using n Points

Last week we looked at counting the maximum number of pieces into which a circle can be cut by n chords (straight lines). Here we will look at a similar-sounding problem where we use all the chords formed by n points on the circle. We’ll also see an important example of why we shouldn’t jump …

## Cutting Up a Circle I: Using n Chords

We’ve looked at how to count diagonals in a polygon; this week and next, I want to consider two different problems (though they look similar at first) dealing with chords of a circle (which are practically the same thing as diagonals of a polygon). In each, what we count will be the regions into which …

## Counting Diagonals of a Polyhedron

(An archive question of the week) In gathering information on how to count the diagonals of a polygon, I found this long discussion about a similar-sounding issue, which is hardly more difficult, yet far more complex. It was interesting to explore what the question means, and take it in different directions, on the way to …

## Polygons and Handshakes

We’ll spend the next couple weeks looking at various counting problems. This topic, called combinatorics, is often studied along with probability, but many of the topics we’ll see here feel more like geometry problems! Here, we’ll be counting the diagonals of a polygon, and handshakes between people at a party. Counting diagonals We’ll start with …

## Counting Divisors – with Conditions

(A new question of the week) Last time we looked at a classic problem for which there is a nice formula, namely counting all divisors of a given number. This time, we will examine a question from last August where we have to count the number of divisors of a specific type, making it more …

## How Many Different Meals Are Possible?

(An archive question of the week) While gathering combinatorics questions, there were several that stood out. This one will serve well to summarize the topic, showing multiple methods for counting, and contrasting other kinds of problems. The problem The question, from 2007, relates to an Arby’s promotion: How Many Different Dinners Can Be Ordered? I …

## Stars and Bars: Counting Ways to Distribute Items

We have been looking at ways to count possibilities (combinatorics), including a couple ways to model a problem using blanks to fill in. Today, we’ll consider a special model called Stars and Bars, which can be particularly useful in certain problems, and yields a couple useful formulas. (I only remember the method, not the formulas.) …

## Permutations and Combinations: Undercounts and Overcounts

(A new question of the week) We have been looking at some combinatorics questions, both easy and challenging. Some questions have come to us in recent weeks that can illustrate how to think your way through relatively difficult problems, including catching errors and interpreting a textbook’s solutions. We’ll see yet again that there are usually …