Puzzle for November 25, 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.
Scratchpad
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Hint #1
In eq.4, replace A with D + E (from eq.3): B + E = D + E + C Subtract E from each side of the above equation: B + E – E = D + E + C – E which becomes B = D + C which may be written as eq.4a) B = C + D
Hint #2
In eq.2, replace C + D with B (from eq.4a): F = B
Hint #3
eq.6 may be written as: E = (B + C + D + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × E = 4 × ((B + C + D + F) ÷ 4) which becomes eq.6a) 4×E = B + C + D + F
Hint #4
In eq.6a, substitute B for C + D (from eq.4a), and B for F: 4×E = B + B + B which becomes 4×E = 3×B Divide both sides of the above equation by 4: 4×E ÷ 4 = 3×B ÷ 4 which makes E = ¾×B
Hint #5
Substitute ¾×B for E, and B for F in eq.5: C + ¾×B = A – ¾×B + B which becomes C + ¾×B = A + ¼×B Subtract ¼×B from each side of the equation above: C + ¾×B – ¼×B = A + ¼×B – ¼×B which becomes eq.5a) C + ½×B = A
Hint #6
Substitute ¾×B for E, and C + ½×B for A (from eq.5a) in eq.4: B + ¾×B = C + ½×B + C which becomes 1¾×B = 2×C + ½×B Subtract ½×B from both sides of the equation above: 1¾×B – ½×B = 2×C + ½×B – ½×B which becomes 1¼×B = 2×C Divide both sides by 2: 1¼×B ÷ 2 = 2×C ÷ 2 which makes ⅝×B = C
Hint #7
Substitute ⅝×B for C in eq.5a: ⅝×B + ½×B = A which makes 1⅛×B = A
Hint #8
Substitute ⅝×B for C in eq.4a: B = ⅝×B + D Subtract ⅝×B from each side of the equation above: B – ⅝×B = ⅝×B + D – ⅝×B which becomes ⅜×B = D
Solution
Substitute 1⅛×B for A, ⅝×B for C, ⅜×B for D, ¾×B for E, and B for F in eq.1: 1⅛×B + B + ⅝×B + ⅜×B + ¾×B + B = 39 which simplifies to 4⅞×B = 39 Divide both sides of the above equation by 4⅞: 4⅞×B ÷ 4⅞ = 39 ÷ 4⅞ which means B = 8 making A = 1⅛×B = 1⅛ × 8 = 9 C = ⅝×B = ⅝ × 8 = 5 D = ⅜×B = ⅜ × 8 = 3 E = ¾×B = ¾ × 8 = 6 F = B = 8 and ABCDEF = 985368