# Proof of checkerboard pieces

• Feb 13th 2007, 07:31 PM
chillerbros17
Proof of checkerboard pieces
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• Feb 13th 2007, 09:25 PM
CaptainBlack
Quote:

Originally Posted by chillerbros17
Attachment 1718

Problem is in attachment.

B) It is imposible as part b has two white squares adjacent, which
never occurs on a real chess board.

RonL
• Feb 13th 2007, 09:35 PM
CaptainBlack
Quote:

Originally Posted by chillerbros17
Attachment 1718

Problem is in attachment.

6. Prove the following theorems. State the method of proof that you are using in parts (a) and (c).

(a):
Thm: The sum of two prime numbers, each greater than 2, is never a prime. Restatement: For all integers p, q, if p is a prime greater than 2 and q is a prime greater than 2, then p + q is not prime.

If p and q are primes greater than 2, then p and q are both odd, and may
be written p=2a+1, q=2b+1, for some positive integers a, and b.

Now suppose p+q is prime, then as it is greater than 2, it must be odd.
But:

p+q=2a+1+2b+1=2(a+b+1).

Thus p+q is divisible by 2, and hence not prime (as p+q !=2), which
is a contradiction, hence our assumption is false, and p+q is not prime.

RonL
• Feb 14th 2007, 07:51 AM
chillerbros17
Attachment 1724

Here's is b with the right pattern. Sorry for the mistake.
• Feb 14th 2007, 12:11 PM
CaptainBlack
Quote:

Originally Posted by chillerbros17
Attachment 1724

Here's is b with the right pattern. Sorry for the mistake.

See the attachment. This shows how to fit two of each shape together to make a 4x4 section of the chess board pattern. Four of these fitted together
will make the entire board.

RonL