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?
Solution to puzzle 12: I am using brute force algorithm here. Let us start from the divisibility of 5. If two cards remain after dealing them for 5, the largest value less than 52 will be 47 If there are 47 cards and they are dealt to 3 people, 2 cards are left If they are dealt to 4 people 47 - 44 = 3 cards are left So 47 is the correct answer. Puzzle 13 : Alibaba's cave This puzzle is from mrbartonmaths Ali Baba found a cave full of gold and diamonds. A bag full of gold weighs 200 kilograms, a bag full of diamonds weighs 40 kilograms. Ali Baba can carry only 100 kilograms at a time. A kilogram of gold costs $20, and a kilogram of diamonds costs $60. What is the greatest amount of money Ali Baba can earn for the gold and diamonds he can carry out at once (in one attempt in one bag)?
On a table, 3 normal playing cards are placed face down. To the right of a king, there is a queen or queens. To the left of a queen there is a queen or queens. To the left of a heart, there is a spade or spades. To the right of a spade, there is a spade or spades. Which three cards are on the table? (Borrowed from the book "The colossal book of short puzzles and problems" by Martin Gardner) Show answer The first rule says that there is atleast one king. Second rule says that there are two queens. (KQQ,QKQ) Third rule says that there is a heart and a spade. And the fourth rule says that there are two spades. (HSS,SHS,SSH) Which means out of three cards, two are queens and one is king. And two are spades and one is hearts. So the four possibilities are KS, QS, QH KS, QH, QS QS, KS, QH QS, KH, QS The fourth is an impossible option as there can't be two queen of spades in a deck. So the three possible answers are King of s...
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