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 xx. And I found something! Here is what I ended up with:

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

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

For example:

  1. Perfect squares starting with 8888:

1028888=9428(94282=88887184)1038888=94277(942772=8888152729)1048888=942762(9427622=88880018864) \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)\\

(so on, n would increase by 1 each time...)

  1. Perfect squares starting with 987654321:

105987654321=3142696806(31426968062=9876543214442601636)106987654321=31426968053(314269680532=987654321004282610809) \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 xx 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
5 years, 2 months ago

No vote yet
1 vote

  Easy Math Editor

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. numbered
2. 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 1

paragraph 2

paragraph 1

paragraph 2

[example link]( 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.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

Have you read the solution to the problem? it should be pretty clear that I'm not just randomly testing for the value of nn.

Calvin Lin Staff - 5 years, 2 months ago

Log in to reply

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

Andrew Tawfeek - 5 years, 2 months ago

Log in to reply

Moving this into the solution discussion of the problem directly.

Calvin Lin Staff - 5 years, 2 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...