BrainDen.com - Brain Teasers
• 0

# An alternating squares "aha" puzzle

Go to solution Solved by kukupai,

## Question

Evaluate 1002 - 992 + 982 - 972 + 962 - 952 + ... + 22 - 12 = ?

## Recommended Posts

• 0
• Solution

There are 50 pairs of n

2 - (n -1)2.
n2 - (n -1)2= n + (n- 1)
So 1002 - 992 + 982 - 972 + 962 - 952 + ... + 22 - 12 can be written
100 + 99 + 98+ ... + 2 + 1 = (100 x 101)/2 = 5050

• 0

5050?

##### Share on other sites
• 0

This could be split into 50 pairs, each given by (2n)^2 - (2n-1)^2 = 4n-1. Summing this between 1 and 50 gives 2n(n+1)-n = 5050. Not an "aha" solution, but I had to write it down

##### Share on other sites
• 0

This could be split into 50 pairs, each given by (2n)^2 - (2n-1)^2 = 4n-1. Summing this between 1 and 50 gives 2n(n+1)-n = 5050. Not an "aha" solution, but I had to write it down

Hi Joe, and welcome to the Den.

Pairing is the right first step. Can you make the pairs even simpler?

##### Share on other sites
• 0

I'm not sure it makes it any easier but you could pair from opposite ends, like in the sum to n.

Each pair is then equal to 101 times the difference between the two terms, summing to 101(100-99+98...+2-1) as the terms become negative for higher odd numbers which should give 101(50) =5050

##### Share on other sites
• 0

Good job all. Several great answers.

##### Share on other sites
• 0

Let S

n be a n x n square with it's sides parallel to coordinate axis and bottom left corner at (0,0).
Let's paint green all unit squares of S100 that stay, and let's paint red all unit squares that are removed.
We can see that S100 is divided into 100 L-shaped areas with alternating colors:
most outer area is green, most inner area (degenerated to just 1 unit square) is red.
Out of two neighboring L-shaped areas, the bigger one has two more unit squares than the smaller one.
We have 50 such red-green pairs, therefore we have 100 green unit squares more than red unit squares.
The area of S100 equals 10000, it's half equals 5000, so there has to be 4950 red squares and 5050 green squares.
Number of green squares (5050) is the answer.
##### Share on other sites
• 0

Let S

n be a n x n square with it's sides parallel to coordinate axis and bottom left corner at (0,0).

Let's paint green all unit squares of S100 that stay, and let's paint red all unit squares that are removed.

We can see that S100 is divided into 100 L-shaped areas with alternating colors:

most outer area is green, most inner area (degenerated to just 1 unit square) is red.

Out of two neighboring L-shaped areas, the bigger one has two more unit squares than the smaller one.

We have 50 such red-green pairs, therefore we have 100 green unit squares more than red unit squares.

The area of S100 equals 10000, it's half equals 5000, so there has to be 4950 red squares and 5050 green squares.

Number of green squares (5050) is the answer.

Nice.

## Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

×   Pasted as rich text.   Paste as plain text instead

Only 75 emoji are allowed.

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.