## A Very Valuable Conjecture

Here is a very valuable conjecture:

The spelling of every whole number shares at least one letter with the spelling of the next whole number.

Which is to say that:

• “one” and “two” both share an “o”
• “two” and “three” both share a “t”
• etc.

Could that possibly be true for every whole number?

If I were starting a course on geometry or a unit on proof or an activity on deductive logic, I would introduce this conjecture very early in the process. Let me explain what I find so very valuable about this conjecture.

Deduction is hard. It’s an abstract mental act that adults find difficult. (See: the van Hiele’s and their levels.) Too often we rush students to that abstract act, rushing them past the lower van Hiele levels, and we ask them to argue deductively about objects that, to them, are also abstract.

I suspect that, to many students, those proof prompts read something like this:

Given that the base bangles are twice the tonnage of the circumwhoozle and the diagonalized matrox is invertible, prove that all altimeters cross the equation at Quito, Ecuador.

The word “prove” is weird. And, unfortunately, so is every other word in the sentence.

So I cherish opportunities to help students argue deductively with concrete objects, which is what we’re working with here, with the spelling of whole numbers. This conjecture also gives students several different angles on the proof act.

You can ask students to find a counterexample, for example, a useful strategy when first interrogating a conjecture.

Once students have tried several different numbers they may satisfy themselves that the conjecture is true. This is one of the naive proof schemes Harel & Sowder observed in the students they studied. When this proof scheme surfaces in conjectures about geometric shapes, it’s challenging to summon up one new shape after another to challenge the student’s proof by example. It’s trivial, by comparison, to summon up one new number after another and ask the student to check her hypothesis again.

At a certain point in this process, likely after you give several numbers in the millions, your students may transform in two ways:

1. They’ll get tired of trying example after example. “Proof by examples means you have to try all the examples,” you can say, giving you both a moment to reflect on the need for a more rigorous proof scheme, like deductive reasoning.
2. They’ll notice that every number in the millions shares an “n” with every other number in the millions. And same for the billions. And same for the trillions. And … same for the hundreds. And so on.

And suddenly we’re on our way to a proof by exhaustion, which is much more rigorous than a proof by example. Nice.

This conjecture also leaves ample room for you and your students to pose follow-up conjectures. Like, “Does it work for all integers, or just whole numbers?”

I saw the conjecture and saw its value immediately. This is a very valuable kind of conjecture, I thought. But I don’t have many of them. Do you have another you can trade?

[via Futility Closet.]

BTW. You’re worse off in at least one way now than before you knew the conjecture was true. Now, when you ask your students, “Could that possibly be true?” you’re going to have to pretend.

Max:

What I like about game strategies is you go from “what seems to work,” to “will this always work,” to “here’s why this does/doesn’t work” pretty seamlessly.

Is it true for any other language with an alphabet? It fails for German (5/6 using umlauts, 7/8 otherwise), French (2/3), and Spanish (7/8).

What letter(s) of the alphabet do(es) not appear in the spelling of the first 999 whole numbers? Prove it.

List all the factors of every number from 1-100. What do you notice? Which numbers have an even number of factors? An odd number? What do you notice about numbers with an odd number of factors? Can you prove which numbers beyond 100 will have an odd number of factors?

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Max

August 3, 2016 - 12:43 pm -

This is cool! Here are a few places I got when searching for my own conjectures, though neither are as good as this one. Maybe you’ll let me trade three okay ones for your one great one.

1) The famous locker problem (student 1 opens all the lockers, student 2 closes every other locker, student 3 changes the state of every third locker, etc.), especially when posed as “what does it look like at the end?” rather than “which lockers are open?” — it’s more about making conjectures than simply proving them, but it seems to generate a lot of conjectures, and finding counter-examples, and finding patterns, and then having to defend your example, and it can be brute forced for 100 lockers but not for 1000 or 1,000,000 lockers.

2) Which is greater: the product of all the hairs on all the heads of all the humans in the world, or the sum of all the hairs on all the heads of all the humans in the world? This one isn’t nearly so nice because there’s a trick to it (it’s from a Car Talk puzzler, after all) and it doesn’t have a very satisfying transition from “we can’t just brute force it.” It does get at the “multiplication is always bigger than addition” misconception, though, which might be nice.

3) Your Facebook (or Snapchat or Twitter or whatever) friends probably have more friends than you do. https://www.washingtonpost.com/business/technology/your-facebook-friends-have-more-friends-than-you/2012/02/03/gIQAuNUlmQ_story.html. It’s a surprising and fun result to talk about, and it’s clear that it needs something besides brute force to help you think about it (though data scientists brute forced it) but it’s not clear that the proof is at all pleasing or interesting the way the sharing one letter with the next number name is pleasing.

The other place I go are game strategies. For example, games of NIM or tic-tac-toe or my favorite from mathpickle.com (https://www.youtube.com/watch?v=Jvjz_ChelmI) in which students take turns picking prime or composite numbers to make products — one team wins if the products aren’t equal, the other if they are. Conjecturing about which team has the advantage and what the winning strategies are leads to very interesting math. What I like about game strategies is you go from “what seems to work,” to “will this always work,” to “here’s why this does/doesn’t work” pretty seamlessly. But, again, unlike your most elegant conjecture, getting to the conjectures takes time with most games.

2. #### Paul Hartzer

August 3, 2016 - 2:18 pm -

Fun extension for your readers (but likely too advanced/irrelevant for math class): Is it true for any other language with an alphabet? It fails for German (5/6 using umlauts, 7/8 otherwise), French (2/3), and Spanish (7/8).

3. #### Chris Klerkx

August 3, 2016 - 3:44 pm -

Conjecture: The spelling of every whole number contains E, O, or X.

Follow-up: Is the conjecture true with a different set of letters?

Meta-conjecture: The number of letters cannot be reduced below 3.

4. #### Clara

August 4, 2016 - 1:01 am -

Opens up another question: how many letters are the same? One for one digit numbers; two for two digit numbers, does it jump for three digit to three, or is there a quadratic or exponential (or some other) patterned increase? Does it hold?

5. #### Lynn CP

August 4, 2016 - 6:20 am -

I’ve also had success with taking elementary number theory questions. For example:
List all the factors of every number from 1-100. What do you notice? Which numbers have an even number of factors? An odd number? What do you notice about numbers with an odd number of factors? Can you prove which numbers beyond 100 will have an odd number of factors?

6. #### Warren Maierhofer

August 4, 2016 - 8:42 am -

I’m wondering, in math is proof by example merely inductive? What would deductive logic look like for this conjecture? I struggle to find one.

7. #### Thomas

August 4, 2016 - 9:28 am -

If a domain of discourse is finite, then an inductive reasoning process can be turned into a deductive proof, and sometimes it’s just one example away. Neat!

8. #### michael Serra

August 4, 2016 - 4:30 pm -

Prove that there is at least one Friday the thirteenth every year.

Similar to your example (and Chris above): What letter(s) of the alphabet do(es) not appear in the spelling of the first 999 whole numbers? Prove it.

9. #### michael Serra

August 4, 2016 - 4:40 pm -

Others in combinatorial geometry that are great explorations and lead to non-traditional proofs:
Find all the pentominoes and prove there are no others.
Ditto: Find all the hexominoes and prove there are no others.

10. #### michael Serra

August 4, 2016 - 4:46 pm -

Draw a pair of lines using both edges of your ruler. Pick up your ruler and do it again with the second pair of lines intersecting the first pair. What is the quadrilateral formed by the intersection of these four lines? Prove it.

11. #### Paul Hartzer

August 4, 2016 - 4:47 pm -

“What letter(s) of the alphabet do(es) not appear in the spelling of the first 999 whole numbers? Prove it.”

That also leads to the debate about whether “a hundred” or “one hundred and one” are “proper” names for 100 and 101, respectively. (I say they are, but a lot of math teachers, particularly in lower grades, apparently insist they’re not.)

12. #### michael Serra

August 4, 2016 - 4:51 pm -

Draw a convex quadrilateral on your paper. Make a copy of it. Draw one diagonal in one copy and draw the other diagonal in the second copy. Cut out the four triangles. Can you arrange the four triangles into a parallelogram region? If so, prove it is a parallelogram. If not, explain why not.

13. #### michael Serra

August 4, 2016 - 4:58 pm -

“one hundred and one” The “and” as in writing a check, represents a decimal point and would not be a whole number.

14. #### michael Serra

August 4, 2016 - 5:06 pm -

Another great non-traditional proof from recreational mathematics is to prove that the 3×3 magic square is unique (not counting rotations and reflections)

15. #### Paul Hartzer

August 4, 2016 - 5:09 pm -

““one hundred and one” The “and” as in writing a check, represents a decimal point and would not be a whole number.”

I know the argument. I disagree with it. “One hundred and one dollars” is perfectly understandable.

16. #### Dan Meyer

August 4, 2016 - 6:09 pm -

Michael:

What letter(s) of the alphabet do(es) not appear in the spelling of the first 999 whole numbers? Prove it.

That’s a beaut. Added to the post.

17. #### Dan Meyer

August 4, 2016 - 6:11 pm -

Lynn CP:

List all the factors of every number from 1-100. What do you notice? Which numbers have an even number of factors? An odd number? What do you notice about numbers with an odd number of factors? Can you prove which numbers beyond 100 will have an odd number of factors?

Nice. Also added to the post.

18. #### Peter Booth

August 5, 2016 - 3:38 am -

Perhaps I’m not thinking this through — which is very possible — but two thoughts occur to me….

Almost every number ends with “and one” or “and two” or “and three” etc. Right? So wouldn’t showing that 1,2,3,4,5,6,7,8 and 9 all fit this pattern suffice for almost every number?

The exception to this are numbers that end in zero, like “One million” or “one hundred forty-two thousand nine hundred and thirty”. Ignoring the fact for the moment that just about the entire number in my example has the same letters as the next one, wouldn’t you run into trouble when you get out to numbers that have not been named? There has to be a biggest number with a name doesn’t there? (Googolplex?) So what happens when you move beyond that number? If we’re talking “proof” wouldn’t you need to account for that? Am I being too demanding of the word Proof? (Not a rhetorical question.) :)

19. #### Paul Hartzer

August 5, 2016 - 4:08 am -

“Almost every number ends with “and one” or “and two” or “and three” etc. Right?”

That depends on the convention. Many math teachers (like Michael Serra) insist that “one hundred and one” means 100.1, or something like that. I disagree, but you’ll meet many students who absolutely insist on that as well, because their teachers scolded them into it.

But that doesn’t matter, because 100, 101, … 999 all have “hundred”; 1000, 1001, … 999,999 all have “thousand”, and so on. Since “hundred”, “thousand”, and “*illion” all have “n”, it suffices to show that 1..100 works. If you’re using the million/million/billion/billiard convention, then “l” becomes the obvious link.

Hey, Chris: “A milliard” doesn’t have any of those letters, so your conjecture doesn’t work in some English naming conventions. ;)

20. #### Chester Draws

August 6, 2016 - 4:13 pm -

Too often we rush students to that abstract act, rushing them past the lower van Hiele levels, and we ask them to argue deductively about objects that, to them, are also abstract.

Isn’t that why we start, traditionally, with proofs in Geometry? They can see the angles in question, so the abstraction is limited.

In introducing proof by Geometry deduction, I wonder how much use a “proof” of this sort is. Geometry proofs don’t use exhaustion, exclusion or induction, so it’s not a great lead in. (I prefer to get the strugglers to do geometry proofs first with some numbers, then repeat the process with variables/pronumerals to lower the mental barriers.)

I can see the proposed “proof” being much more relevant when approaching elementary algebra proofs — where we do actually use exhaustion, induction and exclusion.

So if you were leading a period with this, how much time do you spend on it? Is it a quick taster or a task to fill a period?