Puzzle for November 26, 2019 ( )
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.
Scratchpad
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Hint #1
Add F to each side of eq.5: B + F + F = A – F + F which becomes eq.5a) B + 2×F = A Add B, F, and E to both sides of eq.4: A – B – F + B + F + E = D – E + F + B + F + E which simplifies to eq.4a) A + E = D + B + 2×F
Hint #2
In eq.4a, replace B + 2×F with A (from eq.5a): A + E = D + A Subtract A from each side: A + E – A = D + A – A which makes E = D
Hint #3
In eq.3, replace E with D: D + D = A which makes 2×D = A
Hint #4
In eq.6, substitute 2×D for A: 2×D – B = B – D Add B and D to both sides of the equation above: 2×D – B + B + D = B – D + B + D which means 3×D = 2×B Divide both sides by 2: 3×D ÷ 2 = 2×B ÷ 2 which makes 1½×D = B
Hint #5
Substitute 1½×D for B, and 2×D for A in eq.5: 1½×D + F = 2×D – F In the above equation, subtract 1½×D from both sides, and add F to both sides: 1½×D + F – 1½×D + F = 2×D – F – 1½×D + F which means 2×F = ½×D Divide both sides by 2: 2×F ÷ 2 = ½×D ÷ 2 which makes F = ¼×D
Hint #6
Substitute 1½×D for B, and ¼×D for F in eq.2: C = 1½×D + ¼×D which makes C = 1¾×D
Solution
Substitute 2×D for A, 1½×D for B, 1¾×D for C, D for E, and ¼×D for F in eq.1: 2×D + 1½×D + 1¾×D + D + D + ¼×D = 30 which simplifies to 7½×D = 30 Divide both sides by 7½: 7½×D ÷ 7½ = 30 ÷ 7½ which means D = 4 making A = 2×D = 2 × 4 = 8 B = 1½×D = 1½ × 4 = 6 C = 1¾×D = 1¾ × 4 = 7 E = D = 4 F = ¼×D = ¼ × 4 = 1 and ABCDEF = 876441