PUZZLES
1) Find the next number in the sequence: 10, 25, 39, 77, 679, 6788, …
2) Create a sequence starting with any two random numbers. Add them together to create the third number in the sequence. Add the third and second to create the fourth. Add the fourth and third to create the fifth. And so on. The ratio of consecutive numbers in the sequence (ie the second to the first – whatever they may be – the third to the second, fourth to the third, etc) converges; how quickly can you determine the number to which the ratio converges?
The answers to last month’s puzzles were supplied last month.
Here are the answers to this month’s puzzles:
1) Start with any number X containing two or more digits. Multiply all of X’s digits together. Take the result and multiply its digits together. Continue until your result is a single digit. The amount of times you have to multiply X’s digits to get down to a single digit is the “persistence” of X. Our question this month touches on persistence. The key to our sequence is that 10 is the smallest number with persistence one, while 25 is the smallest with persistence two, and so forth. Ten is the first term, because it’s the smallest number (of two digits or more), which reduces to one digit in one step. 25 is the second term, because it reduces to 10, which reduces to 0. Therefore 25 is the smallest number that reduces to a single digit in two steps. 39 is the third term, because it’s the smallest number that reduces to one digit in three steps (39 becomes 27 which becomes 14 which becomes 4). 77 reduces to one digit in four steps. And so on. The next number after 6788 is 68889 (it’s the seventh term and smallest number that reduces to one digit in seven steps). [from Here’s Looking at Euclid, by Alex Bellos, based on work by Neil Sloane]
2) They converge on the golden mean, phi, which is 1.618…[ from Here’s Looking at Euclid, by Alex Bellos]
