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Math Magic Archive

Problem of the Month
(June 2019)

Several times, the problem of the month has been something to do with Friedman Numbers, whether it be S-Numbers, Self-Reproducing Numbers, or Fractional, Redundant, Almost, and Approximate Friedman Numbers. This month we consider a few other variations.

Problem #1: A number is an anti-Friedman number if it has no repeated digits and it can be formed using one of each digit NOT in the number, together with addition, subtraction, multiplication, division, exponentiation, and concatenation. For example, 592710 = 843 + 6. There are finitely many anti-Friedman numbers, and it might be possible to find them all. What is the largest one you can find? I'm pretty sure all numbers with 3 or fewer distinct digits are anti-Friedman numbers. What are the smallest numbers that are NOT anti-Friedman numbers?

Problem #2: A number is a k-shifted Friedman number if it can be written using its digits shifted by some constant k. The constant k can be positive or negative as long as the shifted digits are all between 0 and 9. For example, with k=1, 108 = 12 × 9. If the shifted digits can be used in order, we call the number a nice k-shifted Friedman number. For example, with k=1, 178 = 2 × 89. What are the small k-shifted Friedman numbers? Which of them are nice? What are the smallest k-shifted Friedman numbers with shifts of –8, –7, –6, –5, +7, and +8?

Problem #3: A pair of numbers is a Friedman pair if the digits of the first can make the second, and the digits of the second can make the first. For example, 27 = 28 – 1 and 128 = 27. If the digits can be used in order, we call the pair a nice Friedman pair. If only one of them can be written with the digits in order, we call it a semi-nice Friedman pair. What are the small Friedman pairs? Which of them are semi-nice or nice? We can also ask for Friedman triplets, 3 numbers which can be formed by the digits of either of the other two.


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