then we add them thus in our dictionary the word cats would be stored in the array cell with index all the other english words would likewise be assigned an array index calculated by this process how well would this workfor the sake of argumentlet' restrict ourselves to -letter words then (remembering that blank is )the first word in the dictionaryawould be coded by + + + + + + + + + = the last potential word in the dictionary would be zzzzzzzzzz (ten zsour code obtained by adding its letters would be thus the total range of word codes is from to unfortunatelythere are , words in the dictionaryso there aren' enough index numbers to go around each array element will need to hold about words ( , divided by clearly this presents problems if we're thinking in terms of our one word-per-array element scheme maybe we could put subarray or linked list of words at each array element howeverthis would seriously degrade the access speed it would be quick to access the array elementbut slow to search through the words to find the one we wanted so our first attempt at converting words to numbers leaves something to be desired too many words have the same index (for examplewastingivetendmoantickbailsdredgeand hundreds of other words add to as cats does we conclude that this approach doesn' discriminate enoughso the resulting array has too few elements we need to spread out the range of possible indices multiply by powers let' try different way to map words to numbers if our array was too small beforelet' make sure it' big enough what would happen if we created an array in which every wordin fact every potential wordfrom to zzzzzzzzzzwas guaranteed to occupy its own unique array elementto do thiswe need to be sure that every character in word contributes in unique way to the final number we'll begin by thinking about an analogous situation with numbers instead of words recall that in an ordinary multi-digit numbereach digit position represents value times as big as the position to its right thus , really means * * * * orwriting the multipliers as powers of