Puzzle for May 31, 2020 ( )
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.
* DE is a 2-digit number (not D×E). DE is an angle expressed in degrees.
Scratchpad
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Hint #1
There are three combinations of integers for D, E, and F that make eq.4 true: 1. DE = 90 and F = 1, makes sine (90) = 1 = sq.rt (1) ÷ 1 2. DE = 30 and F = 4, makes sine (30) = 1/2 = 2/4 = sq.rt (4) ÷ 4 3. DE = 45 and F = 2, makes sine (45) = sq.rt (2) ÷ 2
Hint #2
Check combination 1: F = 1, D = 9, and E = 0 ... In eq.3, replacing F with 1, D with 9, and E with 0 would yield: A – 1 = 9 – 0 Adding 1 to both sides of the equation above would yield: A – 1 + 1 = 9 – 0 + 1 which would make A = 10 Since A must be a one-digit integer, then: A ≠ 10 and, therefore F ≠ 1 and D ≠ 9 and E ≠ 0
Hint #3
Begin checking combination 2: F = 4, D = 3, and E = 0 ... In eq.3, replacing F with 4, D with 3, and E with 0 would yield: A – 4 = 3 – 0 Adding 4 to both sides of the above equation would yield: A – 4 + 4 = 3 – 0 + 4 which would make A = 7
Hint #4
Continue checking combination 2: F = 4, D = 3, and E = 0 ... In eq.2, replacing D with 3, E with 0, and A with 7 would yield: 3 + 0 = 7 + C Subtracting 7 from both sides of the equation above would yield: 3 + 0 – 7 = 7 + C – 7 which would make –4 = C Since C must be non-negative, then: C ≠ –4 and, therefore F ≠ 4 and D ≠ 3 and E ≠ 0 which means F = 2 and D = 4 and E = 5
Hint #5
Substitute 2 for F, 4 for D, and 5 for E in eq.3: A – 2 = 4 – 5 which becomes A – 2 = –1 Add 2 to each side of the equation above: A – 2 + 2 = –1 + 2 which makes A = 1
Hint #6
Substitute 4 for D, 5 for E, and 1 for A in eq.2: 4 + 5 = 1 + C which becomes 9 = 1 + C Subtract 1 from each side of the above equation: 9 – 1 = 1 + C – 1 which makes 8 = C
Solution
Substitute 8 for C, and 2 for F in eq.1: 8 = B + 2 Subtract 2 from both sides of the above equation: 8 – 2 = B + 2 – 2 which becomes 6 = B and ABCDEF = 168452