Dotty Lines

Geometry Level 3

On a 13 by 13 square grid of dots (where all dots are length 1 apart) a line can be drawn from any point to any point. How many different (positive) integer lengths are there for this line?

13 16 14 12

3 solutions

Michael Ng
May 23, 2014

The length of the line has a minimum of 1, and a maximum of $\lfloor\sqrt{12^2+12^2}\rfloor = 16$ . However, after the horizontal lengths from 1 to 12, longer lengths can only be made by forming diagonals. It follows that these lengths hence must be the hypotenuses of Pythagorean Triples. Only 13 in (5,12,13) and 15 (9,12,15) can fit in the 12 by 12 grid, therefore the answer is 14.

Actually, since it is a 12 x 12 grid, the longest length you can make is 11, causing the answer to be 11.

Chris Hambacher - 7 years ago

Thanks for pointing that out; I've changed it to a 13x13 grid.

Michael Ng - 7 years ago

It is a 13x13 grid, so 1 to 12. And then u have 13 from the triangle with sides 5, 12 and 13. And 15 from the triangle 9, 12 and 15.

Gustavo Jambersi - 5 years, 6 months ago

Sorry, the answer is wrong. You can form 12 on any one line. Then use Pythagorian triples to produce other number. 3, 4 and 5. 5, 12 and 13. 9, 12 and 15. So the smallest answer is 15. However, the diagonal 10 (from 6 and 8) can be viewd as different than a straight 10.

Hatim Zaghloul - 7 years ago

Sorry, you might have counted 5 twice? Also, I said different lengths, not different types of lines; hence the only lengths are 1 to 12, 13 and 15 which gives the answer 14.

Michael Ng - 7 years ago

Thank you.

Hatim Zaghloul - 7 years ago

The question is, "How many different integer lengths are there for this line?" The horizontal-or-vertical 10 is a different line from the oblique (diagonal) 10, but they are the same length, even if the LOGO turtle doesn't round the latter off correctly.

Whitney Clark - 6 years, 6 months ago

How does this question even have that high of a rating?!

Nate Ji - 6 years, 11 months ago

I am trying a guess out here . Suppose you connect three dots diagonally here then that length would be sqrt(4+4)=2.82, four dots then the length would be sqrt(9+9) and so on

Neelabh Vikram - 6 years, 6 months ago

Naimish Khara
Jun 27, 2014

total 1 to 13 we can make 13 different line s but and we can also make length of 15(as per Pythagoras triples (9,12,15) which is inclined so total 13+1=14 different lines can be made

On a grid 13x13, we can make 12 different lines... The lenght 13 we got from the triangle 5, 12 and 13.

Gustavo Jambersi - 5 years, 6 months ago

A Former Brilliant Member
May 28, 2014

Counting trivial lines we have $12$ . Trivial lines are either horizontal or vertical. Now the only other integer lines that are not vertical or horizontal are primitive Pythagorean triples with $a$ and $b$ of $a^2+b^2=c^2$ being less then or equal to 13. There are only 2 triples that fit this criteria, namely $(3, 4, 5)$ and $5, 12, 13$ . Therefore our final answer is $\boxed{14}$

Dotty Lines

3 solutions

0 pending reports