hash tables say we want to convert the word cats to number we convert the digits to numbers as shown earlier then we multiply each number by the appropriate power of and add the results * * * * calculating the powers gives * , * * * and multiplying the letter codes times the powers yields , which sums to , this process does indeed generate unique number for every potential word we just calculated -letter word what happens with larger wordsunfortunatelythe range of numbers becomes rather large the largest -letter wordzzzzzzzzzztranslates into * * * * * * * * * * just by itself is more than , , , , so you can see that the sum will be huge an array stored in memory can' possibly have this many elements figure not word not word not word not word actual english word not word not word fira firb firc fird fire firf firg , , , , , , the problem is that this scheme assigns an array element to every potential wordwhether it' an actual english word or not thusthere are cells for aaaaaaaaaaaaaaaaaaabaaaaaaaaacand so onup to zzzzzzzzzz only small fraction of these cells are necessary for real wordsso most array cells are empty this situation is shown in figure , index for every potential word our first scheme--adding the numbers--generated too few indices this latest scheme--adding the numbers times powers of --generates too many