5.8 Wagner-Oke, Anderson-Jussen. All Horses Are The Same Color: Proof by Induction

5.8 Wagner-Oke, Anderson-Jussen. All Horses Are The Same Color: Proof by Induction 67

Conclusion

What makes the three-way duel problem so enticing is the fact that, even in a situation shrouded by human decision-making, mathematics leads us towards the best possible strategy for each player. In a group of three friends (certainly not the best of friends) where each decides to participate in a “truel,” one can rest assured that, however their shooting accuracy, by following the math, they are maximizing their odds at survival by following the outlined strategy, where any deviation is sure never to increase one’s odds of success.

It turns out that the person with the highest chance of winning is not the person with the highest accuracy, but the person with the lowest accuracy, which is Charlie. Because Arnold is the most accurate of the three, he is also the most targeted, as both Barney and Charlie wish to eliminate him in order to face the weaker opponent in the final round. Similarly, Charlie is the least targeted as Arnold and Barney will fire at each other until one of them is eliminated. Since Charlie is guaranteed the first shot at his opponent (whoever he may be), he has at least a 50 percent chance of winning the duel.

The biggest takeaway from this problem is that "survival of the fittest" does not always mean "survival of the most powerful". For example, in a large, free-for-all sports competition, such as dodge-ball, or a heated game of Mario Kart, the one most likely to win is not necessarily the one with the highest skill, but rather the one who is least targeted by the group, slipping by unnoticed to take a final stab at victory when the time is right [1].

Reference [1] Andrew M. Coleman. Game Theory and its Applications. New York, NY: Psychology Press, 2017 (cited on page 67).

5.8 All Horses Are The Same Color: Proof by Induction

By Neil Wagner-Oke and Jack Anderson-Jussen

Abstract

Are all horses the same color? While some people would be quick to say no, mathematicians might be more hesitant to give a quick answer. Using modern tactics of proof by induction, it may be possible to show that all horses ARE the same color. With a keen knowledge of induction, mathematicians proceed cautiously with this statement. On the surface it may seem crazy to think this is true, but with some mathematical reasoning, it can be proven with false logic. This false logic might fly under the radar for some, but with careful analysis we can say beyond a shadow of a doubt, that all horses are NOT the same color. Let’s start by looking at induction itself, and then see how it falls apart in this case of the horses.

Introduction

Have you ever met someone who doesn’t like brownies? We certainly haven’t, and we would bet that nobody in the world dislikes them either. We could start proving this by saying, we like them, my dad likes them, my cousins all like them, and so on and so on. Unfortunately for us, we can’t ask every single person on the planet, and thus it makes it near impossible for us to prove this. We could boil down the population to a sample size of maybe 100 people, which makes the likelihood of them all liking brownies higher but as this isn’t the whole population we can’t prove our statement true beyond a reason of doubt. Problems such as this can be solved using a method of proof known as induction. Induction

Induction is a technique of mathematical proof. The strength of proof by induction is its ability to prove large sequences of statements. To prove by induction you first take an arbitrary number n, and prove that when n is as low as possible the statement is true, we call this the base case. You then assume the statement n = k is true, and using this you can prove that n = k + 1 is true, this is the induction step. The principles behind induction can be boiled down to this simplification: if you want to prove someone can climb an infinite staircase all you need to do is show they can climb the the first (n = 1), they know how to climb from any step (n = k) to the next step (n = k + 1). If you can prove those two things, they can climb infinitely.

The Horses The Proof

For the base case, we look at all possible herds of horses that have only one horse. We know that each of these herds only has one color of horse because they only have one horse, and a horse can only be one color. Now we assume that all herds of size n only contain one color of horse. Finally, we must extend the assumption to all herds of horses that have n + 1 horses. We can do this because we know that the first n horses in a set of n + 1 horses must be the same color by our assumption. Likewise, the last n horses must be the same color. Therefore the first horse which is included in the first set of n horses must be the same color as the rest of the horses. Those

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horses are also the same color as the horse at n + 1 since the last n horses are the same color . This is illustrated in Figure 1.

Figure 1: The general case for the horse problem. What’s Wrong

This all seems reasonable, following the form of proof by induction, but all horses are not the same color. In addition, using the same method we can also prove other false statements about all of a group being the same. A second example being: all numbers have the same value [1]. So, what went wrong? The problem with this proof is that the inductive step we didn’t take into account every possibility. In this case looking at the transition between n = 1 and n = 2 we see that there is no overlap (Figure 2), which is required for the inductive step. This results in the proof not working for any herds larger than one horse. Chapter 5. Mathematics

Figure 2: The case n = 1 where the augment breaks down.

Conclusion

This poor use of proof by induction shows why we need to be careful when writing proofs. In this case the failing of the proof is obvious because we can see that the proof proves a clearly false statement true. However, this is not always the case. While trying to prove statements that are not clearly true or false it is vital to make certain that a proof has no flaws. This fun thought problem gives us a great way to experience induction in a way anybody can think about, regardless of mathematical prowess.

References [1] All horses are the same color. https : / / en . wikipedia.org/wiki/All_horses_are_the_ same_color. Accessed: 2022-02-22. 2022 (cited on page 68). [2] Laura Pennington. Proof by Induction: Steps and Examples. https : / / study . com / academy / lesson / proof - by - induction - steps examples.html. Accessed: 2022-02-22. 2022. [3] Prof. Sormani. Proof by Induction. http : / / comet . lehman . cuny . edu / sormani / teaching / induction . html. Accessed: 202202-22. 2015.