How many 3x5 rectangles will fit in a rectangle 18 x 26 ?

Assuming length is greater than the breadth for both the rectangles (smaller and larger ones), following cases will occur while you try to pack smaller rectangles on the larger one. Let the length of larger rectangle be L, its breadth be B and length and breadth of smaller rectangles be l and b respectively. Case 1. read more

Case 1. Pack the smaller rectangles such that their lengths are parallel to the breadth of the larger rectangle until you fall short of space. Then try the other way round (Length of larger rectangle parallel to Lengths of smaller one) on the available space. read more

In a rectangle m × n there are (m+1) vertical grid lines and (n+1) horizontal grid lines (7 and 5 in the example here). To define any rectangle within the grid, we must choose 2 of each and there are ( (m+1) choose 2 ) × ( (n+1) choose 2 ) ways to do that. For 4 × 6 that gives us (5 choose 2) × (7 choose 2) = 10 * 21 = 210. read more

[math]t_1=2[/math] (either two separate squares or one rectangle) [math]u_1=3[/math] (either we use a horizontal rectangle + a square, a vertical rectangle + a square, or three squares) We will now look at the general case, and we will construct a recurrence relation for the numbers [math]t_n[/math] and [math]u_n[/math]. read more