The president of a 100-member society receives word that the meeting place must be changed, and he needs to inform the rest of the members. He starts a telephone tree: He informs three members, each of whom informs another three members, and so on until all 100 members have received the news. Using this method, what is the greatest number of members who don’t have to make a call?
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It’s easy enough to work this out by counting:
Round #1: 4 members informed (the president plus the 3 members he calls)
Round #2: 13 members informed (the original 4 plus the 3 × 3 additional members)
Round #3: 40 members informed (the 13 from Rounds #1-2, plus 9 × 3 additional members)
Round #4: 100 members informed (the 40 from Rounds #1-3, plus the 60 remaining members, requiring 20 callers)
Altogether, 4 + 9 + 20 = 33 members have to make calls, so 67 do not.
But it’s easier to work backward. After the president has received the news, 99 members remain to be called. That will require 33 callers.
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