Puzzle for May 9, 2021 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* A! is A-factorial.
** BC is a 2-digit number (not B×C). BC is an angle expressed in degrees.
Once again, we extend our thanks to Tom H for sending us an interesting puzzle. Thank you, Tom!
Scratchpad
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Hint #1
In eq.4, the only non-negative two-digit integers that yield an integer value for secant are: B = 0, and C = 0 which makes secant (00) = 1 = F or B = 6, and C = 0 which makes secant (60) = 2 = F In either of these two cases: C = 0
Hint #2
First, check: B = 0, C = 0, and F = 1 ... Substituting 0 for B and C, and 1 for F in eq.1 would yield: A + 0 = 0 + 1 – A which would become A = 1 – A Adding A to both sides of the above equation would yield: A + A = 1 – A + A which would make eq.1a) 2×A = 1
Hint #3
Finish checking: B = 0, C = 0, and F = 1 ... Dividing both sides of eq.1a by 2 would yield: 2×A ÷ 2 = 1 ÷ 2 which would make A = ½ Since A must be an integer, then A ≠ ½ which means B ≠ 0 and F ≠ 1 and therefore makes B = 6 and F = 2
Hint #4
Substitute 0 for C, 6 for B, and 2 for F in eq.1: A + 0 = 6 + 2 – A which becomes A = 8 – A Add A to both sides of the equation above: A + A = 8 – A + A which makes 2×A = 8 Divide both sides by 2: 2×A ÷ 2 = 8 ÷ 2 which makes A = 4
Hint #5
Substitute 4 for A, 6 for B, and 2 for F in eq.3: 4! = 6 + (E × 2) which is equivalent to 4 × 3 × 2 × 1 = 6 + 2×E which becomes 24 = 6 + 2×E Subtract 6 from both sides of the equation above: 24 – 6 = 6 + 2×E – 6 which makes 18 = 2×E Divide both sides by 2: 18 ÷ 2 = 2×E ÷ 2 which makes 9 = E
Solution
Substitute 4 for A, 9 for E, 6 for B, and 2 for F in eq.2: 4 + 9 = 6 + D – (2 – 9) which becomes 13 = 6 + D – 2 + 9 which makes 13 = 13 + D Subtract 13 from each side of the equation above: 13 – 13 = 13 + D – 13 which makes 0 = D and makes ABCDEF = 460092