# Formula for Calculating Squares Starting with n

Hey guys! I was pretty bored today and I happened to have my calculator on me. And for some reason, this problem was on my mind.

So, I started to think about how it gets solved and such and wanted to generalize some formula that could find all the squares starting with $x$. And I found something! Here is what I ended up with:

$\left\lceil { 10 }^{ n }\sqrt { x } \right\rceil$

So, for a given $x$, it would output a number you'd have to square to get a perfect square starting with $x$.

For example:

1. Perfect squares starting with 8888:

$\left\lceil { 10 }^{ 2 }\sqrt { 8888 } \right\rceil =9428\quad ({ 9428 }^{ 2 }=88887184)\\ \left\lceil { 10 }^{ 3 }\sqrt { 8888 } \right\rceil =94277\quad ({ 94277 }^{ 2 }=8888152729)\\ \left\lceil { 10 }^{ 4 }\sqrt { 8888 } \right\rceil =942762\quad ({ 942762 }^{ 2 }=88880018864)\$/extract_itex] (so on, n would increase by 1 each time...) 1. Perfect squares starting with 987654321: $\left\lceil { 10 }^{ 5 }\sqrt { 987654321 } \right\rceil =3142696806\quad ({ 3142696806 }^{ 2 }=9876543214442601636)\\ \left\lceil { 10 }^{ 6 }\sqrt { 987654321 } \right\rceil =31426968053\quad ({ 31426968053 }^{ 2 }=987654321004282610809)$ (so on...) My question is this: See the $x$ in that formula that I stated at the start of this and never went on to define? That's the thing, I don't know how to define it, as in without guessing and checking, I don't know the smallest n for which the result is valid. I know from playing around that n depends in some way on the amount of digits of x and the parity of the x. If x is even, the smallest n for which the formula is correct will be even (vice versa for odd). Also, the larger x is, the larger n seems to have to be in order for it to hold. I've tried many things but I can't seem to find out how to determine the smallest value n needed for the formula to carry out properly. Could anyone provide some insight? It would be much appreciated. :) This is just for fun! Note by Andrew Tawfeek 3 years, 7 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote  # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or \[ ... $ to ensure proper formatting.
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Have you read the solution to the problem? it should be pretty clear that I'm not just randomly testing for the value of $n$.

Staff - 3 years, 7 months ago

I understood slightly the differences between when (referring to the solution there) N is odd/even how it would affect the value of $n$, but I can't seem to understand what decides the lowest value of $n$.

- 3 years, 7 months ago

Moving this into the solution discussion of the problem directly.

Staff - 3 years, 7 months ago