# What are some good, elementary and maybe also interesting proofs by induction?

I am hosting a one-time class/talk on the concept of infinity for some (talented) high-school students. I want to teach them about proof by induction and I want them to do some exercises (you learn math by doing!). I am therefore looking for easy, elementary and maybe also intersting exercises for someone with little to no experience in proving statements. Some examples that came to mind:

• Proving $$n \leq n^2$$.
• Proving $$n! \leq n^n$$.
• Proving that the angle sum in an $$n$$-sided polygon is $$(n-2)180^\circ$$ for $$n \geq2$$.
• Proving Bernoulli's Inequality: $$(1+x)^n \leq 1+nx$$ for all $$x \geq -1$$.

I have looked at the thread Examples of mathematical induction but most if not all of the examples given here I think are too difficult for the audience.

Any inputs are welcome and appreciated! The result and its' induction proof need not be 100% rigorous, the point is to illustrate the induction proof in simple settings.

• For any $k \in \Bbb{Z^+}$, you can derive a formula for $$\sum_{i=1}^n i^k$$ purely by induction. See this answer for an example, where $k = 3$. Apr 27 at 17:24
• Binomial theorem. Apr 27 at 17:26
• some examples in that thread seem pretty simple and interesting, e.g Tower of Hanoi and the tiling problem. I think that formulas for stuff like sum of squares or sum of cubes, while definitely provable using induction, aren't the most illuminating. The two examples above have a pictorial flavor and show that induction is really a statement about "one after the other" (or in the case of Tower of Hanoi, "layer by layer"), rather than just some collection of formulas (which is what I suspect high school students may think math is about). Apr 27 at 17:44
• some other examples may include: proving that if $X$ is a set with $n$ elements (say $n\geq 1$ is an integer), then the power set has $2^n$ elements. Another example is that "if you know something for two, then it's true for finitely many", for instance the triangle inequality $|a+b|\leq |a|+|b|$ generalizes to $|x_1+\dots +x_n|\leq |x_1|+\dots +|x_n|$. A more "fun" example is to consider a rectangular chocolate bar made up of $n$ 1x1 squares; and the question is how many cuts are required to cut it up into 1x1 squares? There are also induction proofs for divisibility Apr 27 at 17:55
• Related: How to really teach induction. Apr 28 at 6:02

My favorite Induction proofs were always the more "real life" proofs. For example, here's one I have always been a fan of-

In a badminton singles tournament, each player played against all the others exactly once and each game had a winner. After all the games, each player listed the names of all the players she defeated as well as the names of all the players defeated by the players defeated by her. For instance, if $$A$$ defeats $$B$$ and $$B$$ defeats $$C$$. then in the list of $$A$$ both $$B$$ and $$C$$ are included. Prove that at least one player listed the names of all other players.

And here's another one-

In a sports tournament of $$n$$ players, each pair of players plays exactly one match against each other. There are no draws. Prove that the players can be arranged in an order $$P_1,P_2,\dots ,P_n$$ such that $$P_i$$ defeats $$P_{i+1}\;∀i=1,2,\dots ,n−1$$

First: I still think you can scrape some fairly simple examples/proofs by induction from that thread that you are linking, e.g tiling with trominoes or Towers of Hanoi. So carefully go through that list and there really are some suitable ones that are not too difficult

If you're doing mix of weak and strong induction, here are some of my favorites:

1. Number of ways $$S_n$$ to go up $$n$$ stairs where you either take $$1$$ or $$2$$ stairs at a time is a Fibonaci number, since for your first step you can either go up $$1$$ stairs, in which case you can do remaining $$n-1$$ steps in $$S_{n-1}$$ ways, or go up $$2$$ stairs, in which case you can do remaining $$n-2$$ steps in $$S_{n-2}$$ ways, so $$S_n = S_{n-1} + S_{n-2}$$, i.e. you're dealing with Fibonacci series. Here's a video. Make sure to first pose this question to your audience and have them struggle with it. E.g. they can figure out answer for $$3$$, $$4$$, and $$5$$ steps ... and then someone may actually start recognizing the Fibonacci pattern ... so then the question is: why is this so, i.e. how can we actually prove that?

2. What is the minimum number of breaks to break up a chocolate bar made of $$m \times n$$ little pieces into those very pieces?

You can first give this problem to your audience, and kind of play with them by suggesting that first making a break in the middle of the bar to get two somewhat evenly sized chunks might be a more efficient strategy than breaking off one column at a time and breaking that into little pieces one by one, since with larger chunks you can cover more individual breaklines with a single break than with smaller chunks ... But of course it doesn't matter how you do it: it will always take $$n-1$$ breaks to get $$n$$ pieces, since each break will only increase the number of chunks by one. And that insight really requires no induction, but you can use strong induction to really drive this home: first break will divide bar with $$n$$ pieces into a chunk with $$m$$ pieces and a chunk with $$n-m$$ pieces. By inductive assumption, the first will take $$m-1$$ to completely break apart, and the second takes $$n-m-1$$ breaks, for a total of $$1 + (m-1) + (n-m-1) = n-1$$ breaks.

1. Euler Tours: you can make an Euler tour of a connected graph that visits each edge (connection) exactly once if each vertex (node) has even degree (even number of edges attached to it). Again, nice thing with this one is that you can first give your audience some concrete examples of graphs (e.g. use Seven bridges of Konigsburg), where some of them can be done but pothers not, so they can get a feel for this problem. And then do the proof by induction on 'size' of graph (so this is really more of a structural induction proof): First, start at an arbitrary node $$a$$ of your graph, and just start going around following connections making sure not to repeat any. Now, first note that when you can't go any further, you must be back at $$a$$ (nice question for audience: why?). OK, so now you have a tour .. but you probably left of some parts of the graph where certain connections were never visited. But note: all those subgraphs must still have property of each node having even degree (again: why?), so by inductive assumption you can do Euler Tour for each of those. And now you can just 'insert' those partial Euler tours into your original partial tour to make the whole thing an Euler Tour of the whole graph.

I like these examples not only because they are visual and interactive, but also because they show that induction is not just weak mathematical induction. I see do many treatments start with weak mathematical induction and sometimes not even move on to other forms of induction, and it’s doing students a disservice because induction is a much more general concept that they should intuitively grasp rather than get lost in the formal details.

If you are looking for some more traditional algebraic weak mathematical induction ones: there is of course the sum of all numbers $$1$$ through $$n$$ ... but I would start with the algebraically somewhat simpler sum of the first $$n$$ odd numbers ... though both of those results have some nice proofs by picture as well which doesn't require any induction at all. Can you tell I like visuals? :P

• The staircase idea is expressed in more at-a-glance visuals with squares and dominoes, as shown, for instance, at the beginning of youtu.be/Ct7oltmdJrM Apr 27 at 20:16
• @MarkS. Thanks for the link! Apr 27 at 23:28

When I first studied Proof by induction in highschool, the very simple but interesting proof of $$\sum_{i=1}^ni = \frac{n(n+1)}{2}$$ was presented to me. I thought this to be very intuitive and quite straightforward. I believe this is quite well suited for your audience.

• Ah yes, this is an obvious one i missed, thanks! I think this has just the right level for them. Apr 27 at 17:52
• This is a good one, and you can precede it by computing the sum for small $n$ and guessing the formula before proving it via induction. Then it illustrates the nature of some inductive proofs, that they are sometimes able to verify a formula is correct without seeming to give a good explanation of "why" it's true, since the formula was only a guess and didn't come up "naturally" in the course of the proof. (of course, there are many other ways of proving this that are more explanatory.) Apr 27 at 18:25
• A nice variation on this is the sum of odd numbers that is a complete square. Kids like it :) Apr 28 at 5:41

A few problems I often use when teaching induction:

1. You have a collection of $$3^n$$ coins and a balance. All coins except one are genuine, and one of them weighs more than the rest. Prove you can always find which one is counterfeit using only $$n$$ weighings on the balance.
2. Now do this where you have $$3^n - 1$$ coins, where either there is a single heavier counterfeit or all coins are the same weight, and the goal is to determine which is the case. You again have $$n$$ weighings.
3. You have a $$1 \times n$$ chocolate bar. You eat the bar by breaking off a piece from the left of whatever size you’d like, eating it, and repeating until the bar is fully eaten. How many different ways can you eat the chocolate bar? (It’s $$2^n$$.)
4. You have a “jumbled inequality” consisting of $$n \ge 2$$ blanks with inequality signs < and > interspersed, such as _ < _ > _ > _ < _. The goal is to fill the blanks in so that each consecutive pair of numbers is properly related. Prove you can always fill in $$n$$ blanks with the numbers 1, 2, …, $$n$$, using each of those numbers exactly once.
5. You have a $$2^n \times 2^n$$ checkerboard with exactly one square removed. Prove you can cover the rest with L-shaped triominoes without overlap.
6. Prove that all rational numbers can be written as continued fractions.
7. Prove that all rational numbers can be written as Egyptian fractions.
8. You can make a hexagonal pyramid of cans as follows. The top layer has one can. The second has seven arranged in a hexagon with a side length of one. The third has 19 in a hexagon with a side length of two. Prove that a tower of $$n$$ layers has exactly $$n^3$$ cans.

Hope this helps!

All odd squares have residue $$1\bmod 8$$. For $$(2×0+1)^2=1$$ and

$$(2(k+1)-1)^2-(2k+1)^2=8(k+1).$$

This result impacts the factorization of $$2$$ into prime ideals in quadratic domains, and therefore the class field of said domains. It also makes binary representation a relatively good "filter" for identifying squares, for we end up with five-sixth of terminal-bit patterns never corresponding to squares. Other prime bases do little better than one-half.

• This can just be proved by checking all cases mod 8 right? I mean you can prove it with induction but idk if it is the easiest way if you have basic results of multiplication in modular arithmetic.
– qwr
Apr 28 at 3:57

you can prove, by induction, Legendre's theorem on the highest power of a prime $$p$$ that divides $$n!$$ for some $$n \geq p.$$ I answered with that, once, let me go look

Understanding the proof of a formula for $p^e\Vert n!$