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Applying the steps below, you will generate a number, on the condition that every digit of a number produced in each step is different from its other digits.

1. Write down a number with one, two or three digits.

2. Delete at most three consecutive digits and in the places of the deleted digits place the square of the number formed by these digits.

3. For the number you get, repeat steps two and three. If you cannot get a number that satisfies the conditions, stop.

What is the largest number that can be generated through this mechanism?

Example: 307, 3(07), 349, 3(4)9, 3169, ...

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Here's one way to get there:

387

38(7)

3849

38(49)

382401

(3)82401

982401

98(24)01

9857601

98(5)7601

98257601

98(2)57601

98457601

984(5)7601

984257601

984257(6)01

9842573601

Bonus: What is the answer using hex?

Yes, after I got that answer by hand I wrote a program to see if it was the

largest possible. I was amazed to find out that it was. The funny thing

was that it only took me about 5 minutes to do it by hand, whereas it took

the better part of a half hour to program it. I started with 987 and began

by squaring the 7. From then on, things went pretty smoothly and I got to

the answer. Then, I noticed that I had 2 9s in one of the steps. Then I

realized that I could have started with 387 and made the 3 into a 9 at the

appropriate later step. This is probably the first time I ever maximized

something this complicated without a program!

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