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Yellow stars with the faces of Martin Gardner and Salvador Dali next to text reading “He wrote it! He solved it!” In a pale red box, yellow text reads: “Now it’s your turn to try...The Persistence Problem”

Can You Solve a Puzzle by Martin Gardner, the Best American Puzzler of All-Time?

by

Challenge yourself and your friends with this fun number puzzle.

Martin Gardner was one of the most important American puzzlers and popular science writers ever. He lived from 1914-2010, and wrote thousands of fascinating puzzles in his lifetime, as well as writing a deeply important popular science column weekly for 25 years.

Some of the best minds of all time were challenged and delighted by Martin Gardner’s puzzles. Some of his famous fans included W.H. Auden, Salvador Dalí, and Carl Sagan. The Spanish surrealist artist Salvador Dalí was not only an admirer, friend, and puzzle fan, but he consulted with Gardner about the four-dimensional hypercubes that he incorporated into his art.

One of Gardner’s most beloved puzzles is the persistence puzzle. It’s pretty tricky. Let’s find out if you are up to the challenge!

First, we need to define persistence. A number’s multiplicative persistence is the number of times you need to multiply the digits together to produce a one-digit number.

For example, 27 has a persistence of two because it requires two multiplications to reduce it to one digit. 27 is made up of the digits 2 and 7. If multiply 2×7 the answer is of course 14, which is still a two-digit number. What are the digits of 14? 1 and 4. When you multiply 1×4, the answer is 4. We arrived at a one digit number. We’re done! How many multiplication steps did it take to get 27 to a one-digit number (i.e., to 4)? 2 steps (2×7 and 1×4) so we say that 27 has a persistence of 2. (27 >> (2×7) >> 14 >> (1×4) >> 4)

Let’s try some persistence problems.

What is the smallest number of persistence one?

The answer is 10.

10 >> (1×0) >> 0

We did one multiplication problem to get to a one digit number. One multiplication problem means 10 has a persistence of 1.


What is the smallest number with a persistence of two?

The answer is 25.

25 >> (2×5) >> 10 >> (1×0) >> 0

We did two multiplication problems to get to a one digit number. Two multiplication problems means 25 has a persistence of 2.


Now let’s try to find the smallest number with a persistence of three.

The answer is 39.

39 >> (3*9) >> 27 >> (2×7) >> 14 >> (1×4) >> 4

We did three multiplication problems to get to a one digit number. Three multiplication problems means 39 has a persistence of 3.



Here is a harder one for you. Can you figure out the smallest number with a persistence of four?

Try to solve it, then click to see the answer!

Five yellow stars with red numbers on them on a red background. Pale yellow text reads “77 is the smallest number with a persistence of four. Did you solve it?”

Do you want to learn more about multiplicative persistence? Wolfram Mathworld has a great explanation.

IMACS is different. Read what our graduates say to find out why.

If your child thinks multiplicative persistence is interesting, check out this Martin Gardner book of puzzles and try a free IMACS class. Discover how much fun math can be!