Puzzle for March 24, 2022 ( )
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 D and F to both sides of eq.3: C – D + D + F = B – F + D + F which becomes eq.3a) C + F = B + D Add A to both sides of eq.6: A + C + F + A = B + D + E – A + A which becomes eq.6a) 2×A + C + F = B + D + E
Hint #2
In eq.6a, replace C + F with B + D (from eq.3a): 2×A + B + D = B + D + E Subtract B and D from each side of the equation above: 2×A + B + D – B – D = B + D + E – B – D which simplifies to 2×A = E
Hint #3
In eq.4, replace E with 2×A: 2×A – D = A + D In the above equation, subtract A from both sides, and add D to both sides: 2×A – D – A + D = A + D – A + D which makes A = 2×D Divide both sides by 2: A ÷ 2 = 2×D ÷ 2 which makes ½×A = D
Hint #4
In eq.3, substitute ½×A for D: C – ½×A = B – F In the above equation, add ½×A to both sides, and subtract B from both sides: C – ½×A + ½×A – B = B – F + ½×A – B which becomes C – B = –F + ½×A which is the same as eq.3b) C – B = ½×A – F
Hint #5
Substitute 2×A for E in eq.2: eq.2a) B + C = 2×A + F
Hint #6
Add the left and right sides of eq.3b to the left and right sides of eq.2a, respectively: B + C + C – B = 2×A + F + ½×A – F which becomes 2×C = 2½×A Divide both sides of the above equation by 2: 2×C ÷ 2 = 2½×A ÷ 2 which makes C = 1¼×A
Hint #7
Substitute 1¼×A for C in eq.5: B – 1¼×A = A + 1¼×A – B which becomes B – 1¼×A = 2¼×A – B Add 1¼×A and B to both sides of the equation above: B – 1¼×A + 1¼×A + B = 2¼×A – B + 1¼×A + B which makes 2×B = 3½×A Divide both sides by 2: 2×B ÷ 2 = 3½×A ÷ 2 which makes B = 1¾×A
Hint #8
Substitute 1¾A for B, and 1¼×A for C in eq.2a: 1¾A + 1¼×A = 2×A + F which becomes 3×A = 2×A + F Subtract 2×A from each side of the equation above: 3×A – 2×A = 2×A + F – 2×A which makes A = F
Solution
Substitute 1¾×A for B, 1¼×A for C, ½×A for D, 2×A for E, and A for F in eq.1: A + 1¾×A + 1¼×A + ½×A + 2×A + A = 30 which simplifies to 7½×A = 30 Divide both sides of the above equation by 7½: 7½×A ÷ 7½ = 30 ÷ 7½ which means A = 4 making B = 1¾×A = 1¾ × 4 = 7 C = 1¼×A = 1¼ × 4 = 5 D = ½×A = ½ × 4 = 2 E = 2×A = 2 × 4 = 8 F = A = 4 and ABCDEF = 475284