Four fours from zero to ten

Question

Write each of the numbers 0, 1, 2, ..., 10 using exactly four 4s, combined with addition, subtraction, multiplication, division and brackets. Joining digits to make 44 is allowed; square roots and factorials are not needed for 0 to 10 but may be used if you like. Which of the eleven numbers is the hardest, and why?

Accepted Answer

One solution set, using only the four basic operations: - 0 = 44 minus 44 - 1 = 44 / 44 - 2 = 4/4 + 4/4 - 3 = (4 + 4 + 4) / 4 - 4 = 4 + (4 minus 4) × 4 - 5 = (4 × 4 + 4) / 4 - 6 = (4 + 4) / 4 + 4 - 7 = 44 / 4 minus 4 - 8 = 4 + 4 + 4 minus 4 - 9 = 4 + 4 + 4/4 - 10 = (44 minus 4) / 4 Each line uses exactly four 4s. Which is hardest. With only the four basic operations and no concatenation, 10 and 7 are the awkward ones; allowing 44 makes them easy (7 = 44/4 minus 4 and 10 = (44 minus 4)/4). With square roots and factorials permitted (root 4 = 2, 4! = 24), every number up to well over a hundred …

Four fours from zero to ten

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