An artist has a strange idea that a canvas looks good only if its perimeter is equal to its area. Now the question is how many such rectangles are there with integer sides - rectangles with their perimeter equal to their area? And why?
Solution :
Area = l * w
Perimeter = 2l+2w
area = perimter => l*w = 2l+2w Moving 2l+2w to left hand side, we get 2l+2w+l*w = 0
Adding 4 to both sides we get=> -2l-2w + lw+4 = 4
=> (l-2)(w-2) = 4
The integer pairs with 4 as product are (1,4),(2,2),(4-1)
If l-2 = 1 and w-2=4, we get l = 3 and w = 6
If l-2 = 2 and w-2 = 2, we get l = 4 and w = 4
If l-2 = 4 and w-2 = 1, we again get l = 6 and w = 4
Solutions 1 and 3 are identical.
So the only two rectangles with integers sides and area and perimeter being equal are rectangles with sides 3 and 6 and 4 and 4.
Solution to puzzle 8: Maria is not counting Sundays. So she is not counting 1/7th of days of her age. That is to say she is only counting 6/7th of her age. 6/7th of her age = 30 Her actual age = 30 X 7/6 = 210/6 = 35 years. Puzzle 9 : The 4 coins problem You’re creating a new coin system for your country. You must use only four coin values and you must be able to create the values 1 through 10 using one coin at a minimum and two coins maximum. What 4 coins do you choose, and can you think of a second set of 4 coins that achieves the same goal?
10 prisoners share a cell. The prison warden tells them that the next day they’ll be randomly lined up for execution. He’ll place either a black or white cap on everyone’s head. Each prisoner starting from the back of the line will guess what color his hat is. If he gets it wrong he’ll be executed. Every prisoner can not see the hat on his head or of those behind him. He can see the hats of everyone in front of him. How can the prisoners devise a strategy to save their lives? You can find the solution here .
Image from: Popular Mechanics By now you might have heard many, many, many puzzles of some people always telling truth, and others always telling lies. But in real life, we all neither knights, nor knaves. Anyway this puzzle is slightly different - there are couples and these couples both lie or both tell truth or one lies and other one tells truth. (We all like to believe ourselves to be the last one - wink wink) Dags, Eggs and Fens are being interrogated as one of them is a spy. One of the couples always tells truth. In another couple, both husband and wife always lie. And in the third couple, one of the spouses always lies and the other one always tells truth. Here are the statements by the three couples when interrogated. Mr. Dag: I am not the spy Mrs. Dag: Mr. Egg is the spy. Mr. Egg: Mr. Dag is truthful. Mrs. Egg: Mr. Fen is the spy. Mr. Fen: I am not the spy. Mrs. Fen: Mr. Dag is the spy. Which person is a spy? (This puzzle is take...
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