I'm working on a word problem that I can't seem to wrap my head around. Can someone please breakdown how to solve this? Thanks!
How many square regions are there on a chess board ( a square region can have sides of lenght one through eight) ?
What about on an n X n chess board?
CHESS BOARD
This topic has expert replies
- candrapetra
- Newbie | Next Rank: 10 Posts
- Posts: 5
- Joined: Sat Aug 20, 2011 4:28 pm
- sumgb
- Senior | Next Rank: 100 Posts
- Posts: 64
- Joined: Sun Jul 24, 2011 9:11 am
- Thanked: 13 times
- GMAT Score:610
There is a formula to calculate the no of squares here...
if there are n rows and n columns on a board then no. of squares = n^2 + (n-1)^2 + (n-2)^2 + ...+ 2^2 + 1^2
so for a 4 x 4 board there are 16 + 9 + 4 + 1 = 30 squares
similarly for 3 x 3 board there are 9 + 4 + 1 = 14 squares
hope this helps...
if there are n rows and n columns on a board then no. of squares = n^2 + (n-1)^2 + (n-2)^2 + ...+ 2^2 + 1^2
so for a 4 x 4 board there are 16 + 9 + 4 + 1 = 30 squares
similarly for 3 x 3 board there are 9 + 4 + 1 = 14 squares
hope this helps...
- gmatboost
- Master | Next Rank: 500 Posts
- Posts: 312
- Joined: Tue Aug 02, 2011 3:16 pm
- Location: New York City
- Thanked: 130 times
- Followed by:33 members
- GMAT Score:780
You don't need to memorize a formula. Instead, reason through it:
Let's use an 8x8 board.
How many 8x8 squares are there? 1
How many 7x7 squares are there?
Well, start with one at the top left. We could make new ones by moving it:
down 1 unit
over 1 unit
down and over 1 unit
Total: 4
In other words, the top-left corner of the 7x7 can be in any of the 4 squares defined by a little 2x2 square.
Similarly, for 6x6 squares, the top left corner can be in any of the 9 squares defined by a little 3x3 square.
You can see the pattern emerging: 1 + 4 + 9 ...
Let's use an 8x8 board.
How many 8x8 squares are there? 1
How many 7x7 squares are there?
Well, start with one at the top left. We could make new ones by moving it:
down 1 unit
over 1 unit
down and over 1 unit
Total: 4
In other words, the top-left corner of the 7x7 can be in any of the 4 squares defined by a little 2x2 square.
Similarly, for 6x6 squares, the top left corner can be in any of the 9 squares defined by a little 3x3 square.
You can see the pattern emerging: 1 + 4 + 9 ...
Greg Michnikov, Founder of GMAT Boost
GMAT Boost offers 250+ challenging GMAT Math practice questions, each with a thorough video explanation, and 100+ GMAT Math video tips, each 90 seconds or less.
It's a total of 20+ hours of expert instruction for an introductory price of just $10.
View sample questions and tips without signing up, or sign up now for full access.
Also, check out the most useful GMAT Math blog on the internet here.
GMAT Boost offers 250+ challenging GMAT Math practice questions, each with a thorough video explanation, and 100+ GMAT Math video tips, each 90 seconds or less.
It's a total of 20+ hours of expert instruction for an introductory price of just $10.
View sample questions and tips without signing up, or sign up now for full access.
Also, check out the most useful GMAT Math blog on the internet here.