Maths Puzzles

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?  Show answer 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.  

Let us say the side of equilateral triangle is x.  Then its perimeter is 3x. If the side of hexagon is y, its perimeter is 6y. 3x=6y  ---------since perimeters are same y = 1/2x Area of triangle is 2 2 = (√3/4)x² Area of hexagon= (3√3 y 2 )/2   Replacing y with x/2, we get Area of hexagon =  (3√3 (x/2) 2 )/2 = ( 3√3 x²/4)/2 = 3/2 *  ( √3 x²/4) = 3/2 * 2 = 3 units

An equilateral triangle and a regular hexagon have perimeters of the same length. If the triangle has an area of two square units, what is the area of the hexagon?

Do you need a change of pattern, dear reader? Here is a geometrical puzzle.  Draw 6 line segments of equal lengths to form 8 equilateral triangles. Hint: The triangles need not be of the same size. P.S. Well, the source of this puzzle and many others is this book by T he colossal book of short puzzles and problems by Marin Gardner  and is really wonderful book. (It has solutions too ;) )     Find the solution here. 

 This time we have two geometry problems. And both are images, easy for the eyes and brain.  Find the areas marked with ? in these two puzzles.

Here is non-tricky question - quite simple one ​ An accurate clock shows 8 o'clock in the morning. Through how may degrees will the hour hand rotate when the clock shows 2 o'clock in the afternoon? i) 144º iii) 150º iii) 168º iv) 180º