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original mathematical recreations. If you have a math puzzle, discovery, or observation, please e-mail me about it. You can also send answers to the problem of the month. |

Math Magic Archive |

(June 2019)

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 = 84^{3} + 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 = 2^{8} – 1 and 128 = 2^{7}. 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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