triangular number is the sum of itself and all numbers smaller than itself (number means integer in this context for examplethe triangular number of is because + + + the factorial of number is the product of itself and all numbers smaller than itself for examplethe factorial of is * * * both triangular numbers and factorials can be calculated using either recursive method or simple loop the anagram of word (all possible combinations of its letterscan be found recursively by repeatedly rotating all its letters and anagramming the rightmost - of them binary search can be carried out recursively by checking which half of sorted range the search key is inand then doing the same thing with that half the towers of hanoi puzzle consists of three towers and an arbitrary number of rings the towers of hanoi puzzle can be solved recursively by moving all but the bottom disk of subtree to an intermediate towermoving the bottom disk to the destination towerand finally moving the subtree to the destination merging two sorted arrays means to create third array that contains all the elements from both arrays in sorted order in mergesort -element subarrays of larger array are merged into -element subarrays -element subarrays are merged into -element subarraysand so on until the entire array is sorted mergesort requires ( *logntime mergesort requires workspace equal in size to the original array for triangular numbersfactorialsanagramsand the binary searchthe recursive method contains only one call to itself (there are two shown in the code for the binary searchbut only one is used on any given pass through the method' code for the towers of hanoi and mergesortthe recursive method contains two calls to itself any operation that can be carried out with recursion can be carried out with stack recursive approach may be inefficient if soit can sometimes be replaced with simple loop or stack-based approach