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## Project Euler 206 Solution ## Project Euler 206: Find the unique positive integer for a concealed square

#### Problem Description

Find the unique positive integer whose square has the form 1_2_3_4_5_6_7_8_9_0,
where each “_” is a single digit.

#### Analysis

The square of any number, n, will end in 00 if the last digit of n is zero.  So, our square root ends in a zero and the square has the form  1_2_3_4_5_6_7_8_900.

We can reduce our search to 1_2_3_4_5_6_7_8_9 and multiply the square root (when we find it) by 10.  Now, the only way to end a square with 9 is to square a number ending with 3 or 7. OK, so our square root ends with 3 or 7, an odd number, and we can skip even-numbered n from consideration.

Let’s begin our search with the maximum possible number, 19293949596979899 and take its square root. Starting with an odd number we count down by 2 until the square fits our pattern.

#### Project Euler 206 Solution

Runs < 0.001 seconds in Python 2.7. Use this link to get the Project Euler 206 Solution Python 2.7 source.

#### Afterthoughts

• The square root can only be a 9 digit integer that starts with 1 and ends with 0.
Project Euler 206 Solution last updated

## Discussion

### 7 Responses to “Project Euler 206 Solution”

1. I do like your approach to matching. I came here because I couldn’t see what was wrong with my solution. As it happens, I was offering the square rather than the square root. 🙁

Posted by Bill | March 19, 2016, 10:26 AM
• Posted by Mike Molony | March 23, 2016, 11:40 PM
2. The assumption made is incompletely justified for the problem statement

The question asks for the number whose square is 1_2_3…_9_0

There is no explicit guarantee that n ends in 0, only that n^2 ends in 0.

A better rationale is that there is no value other than a multiple of 10, that when multiplied by itself which gives x0, thus n must be divisible by 10

Posted by Philip Whitehouse | December 8, 2012, 6:01 PM
• Phil, if someone planted an explosive device under your chair and told you that to disarm it all you had to do was type in any integer ending with a zero that, when squared, would not end with a zero would you be blown left or right from the concussion?

Posted by Mike | February 3, 2014, 11:25 PM
• PHIL GOT PWNT!!!!

Posted by George | August 24, 2015, 7:29 PM
• If a proper square ends in 0, then its square root ends in a zero.
Proof:
0x0=
1×1=
2×2=
3×3=
4×4=1
5×5=2
6×6=3
7×7=4
8×8=6
9×9=8
Occurances of a square ending in a zero: 1; 0x0.
So, the square root ends in a zero.

Posted by Anonymous | July 28, 2017, 12:59 PM
• I forgot the =0 in 0x0=0. whoops

Posted by Anonymous | July 28, 2017, 1:00 PM