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.

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?

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.

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.

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.

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.

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

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.) :)

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?