Puzzle for April 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.
Scratchpad
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Hint #1
Subtract the left and right sides of eq.3 from the left and right sides of eq.5, respectively: A + E – (A + B) = B + C + D – (C + E) which becomes A + E – A – B = B + C + D – C – E which becomes E – B = B + D – E Add B and E to both sides of the above equation: E – B + B + E = B + D – E + B + E which becomes eq.5a) 2×E = 2×B + D
Hint #2
Subtract F from both sides of eq.2: A – F = B + D + F – F which becomes eq.2a) A – F = B + D In eq.4, add E to both sides, and subtract F from both sides: A + D – E + E – F = E + F + E – F which becomes A + D – F = 2×E which may be written as eq.4a) A – F + D = 2×E
Hint #3
In eq.4a, replace A – F with B + D (from eq.2a), and 2×E with 2×B + D (from eq.5a): B + D + D = 2×B + D which becomes B + 2×D = 2×B + D Subtract B and D from both sides of the equation above: B + 2×D – B – D = 2×B + D – B – D which makes D = B
Hint #4
In eq.5a, substitute B for D: 2×E = 2×B + B which makes 2×E = 3×B Divide both sides of the equation above by 2: 2×E ÷ 2 = 3×B ÷ 2 which makes E = 1½×B
Hint #5
Substitute 1½×B for E, and B for D in eq.5: A + 1½×B = B + C + B which becomes A + 1½×B = 2×B + C Subtract 1½×B from both sides of the equation above: A + 1½×B – 1½×B = 2×B + C – 1½×B which becomes eq.5b) A = ½×B + C
Hint #6
In eq.6, substitute ½×B + C for A (from eq.5b), and B for D: ½×B + C + F = B + C – F In the above equation, subtract ½×B and C from both sides, and add F to both sides: ½×B + C + F – ½×B – C + F = B + C – F – ½×B – C + F which simplifies to 2×F = ½×B Divide both sides by 2: 2×F ÷ 2 = ½×B ÷ 2 which makes F = ¼×B
Hint #7
Substitute B for D, and ¼×B for F in eq.2: A = B + B + ¼×B which makes A = 2¼×B
Hint #8
Substitute 2¼×B for A in eq.5b: 2¼×B = ½×B + C Subtract ½×B from both sides of the above equation: 2¼×B – ½×B = ½×B + C – ½×B which makes 1¾×B = C
Solution
Substitute 2¼×B for A, 1¾×B for C, B for D, 1½×B for E, and ¼×B for F in eq.1: 2¼×B + B + 1¾×B + B + 1½×B + ¼×B = 31 which simplifies to 7¾×B = 31 Divide both sides of the above equation by 7¾: 7¾×B ÷ 7¾ = 31 ÷ 7¾ which means B = 4 making A = 2¼×B = 2¼ × 4 = 9 C = 1¾×B = 1¾ × 4 = 7 D = B = 4 E = 1½×B = 1½ × 4 = 6 F = ¼×B = ¼ × 4 = 1 and ABCDEF = 947461