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Incubator Q&A

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Comments on Droppable Square numbers

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Droppable Square numbers Question

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A number $\large X$ is defined to be a Droppable Square number if the ordinary base-10 representation of $\large X^2$ can be formed by inserting the digit $\large 2$ anywhere in the base-10 representation of $\large X$, even at the beginning or end of $\large X.$

Rearranging the digits of $\large X$ at any time is not allowed.


$\large 11$ is a Droppable Square number because $\large 11^2 \ = \ 121$ can be formed by inserting the digit $\large 2$ into the middle of $\large 11$ to form $\large 121.$

$\large 5$ is also a Droppable Square number because $\large 5^2 \ = \ 25$ can be formed by inserting the digit $\large 2$ at the beginning of $\large 5$ to form $\large 25.$


Find another Droppable Square number.


History

2 comment threads

[SPOILERS] Oh that's very clever. I see what you did there. :-) (13 comments)
Spoiler thread (3 comments)
[SPOILERS] Oh that's very clever. I see what you did there. :-)
Monica Cellio‭ wrote 4 months ago

Oh that's very clever. I see what you did there. :-)

Skipping 3 deleted comments.

trichoplax‭ wrote 4 months ago

Are you thinking of posting an answer? Sounds like you had an insight before I did.

Monica Cellio‭ wrote 4 months ago

I am. I wanted to give others a chance first, since it seems people don't tend to try after there's a correct answer. In this case I believe there are at least two approaches to answering, so I'll probably post one soon.

trichoplax‭ wrote 4 months ago

I suppose people might try without looking at an existing answer, but then not post anything when they check and see that the existing answer is the same as their own. That's one reason I recently tried making a puzzle with a leaderboard rather than a single correct answer.

I only have 1 approach in mind for an answer here (although it would give 2 answers). I'm curious what your 2 approaches are now, and whether mine is one of them. I'll avoid spoilers in this thread though. I'm also happy to wait and see if anyone else answers. It would be good to see answers from different people.

Monica Cellio‭ wrote 4 months ago

I have one approach that produces two answers, and I suspect it's the one you're thinking of. I'm aware of another approach but I'm not sure how to apply it to actually produce an answer; I just believe it is theoretically possible.

Monica Cellio‭ and trichoplax‭ I have posted an answer.

Monica Cellio‭ wrote 4 months ago

I posted an answer after seeing that Will had done so. @trichoplax‭ I hope you'll go ahead and post yours; even if you came up with some same answers, your reasoning might have been different!

trichoplax‭ wrote 4 months ago · edited 4 months ago

Monica Cellio‭ mine were exactly the same as yours so I don't think it would add anything to post an answer. I like that yours comes with an interesting explanation. Since I didn't have an explanation for mine, my answer would just be the numbers...

trichoplax‭ wrote 4 months ago

will.octagon.gibson‭ I'd convinced myself that the answers I came up with (identical to Monica's) were the only ones possible, so it was really interesting to see your completely different approach. This challenge had more layers than I expected.

Monica Cellio‭ wrote 4 months ago

Irrational numbers never occurred to me until I saw how will.octagon.gibson‭ approached it. And even with that clue, I had to scrape off a lot of cobwebs for the math to find them, so I'm really glad for the detailed explanation.

trichoplax‭ You are not the only one who didn’t see all the solutions. I didn’t discover the 1.1 and .5 solutions, but Monica did.

Monica Cellio‭ I agree that my answer wouldn’t have been useful without a detailed explanation of how I discovered that irrational number. By the way, there are more irrational numbers that are Droppable Square numbers.

System‭ wrote 4 months ago

Thread renamed from "Oh that's very clever. I see what you did there. :-)" to "[SPOILERS] Oh that's very clever. I see what you did there. :-)" by trichoplax‭