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Problem of the Month (August 2016)

A flash card is a string of digits or arithmetic operations (+ – × /). We allow unary minuses, but no other leading or trailing operations, or leading zeroes. What is the shortest flash card that when repeated multiple times, evaluates to n? For example, the flash card 6+1 repeated twice looks like 6+16+1, which evaluates to 23. Do flash cards like this always exist?

What are the shortest pairs of flash cards that evaluate to n in one order, and n+1 in the other order? What are the 3 shortest flash cards that have different cyclical orders to evaluate to n, n+1, and n+2?

For pairs A and B of flash cards, is it possible that the 4 orders AA, AB, BA, and BB evaluate to 4 consecutive integers? Is this sort of thing possible with larger numbers of flash cards?


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