Puzzle for April 6, 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.
Scratchpad
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Hint #1
In eq.4, replace F with A + B (from eq.2): B + C = A + A + B which becomes B + C = 2×A + B Subtract B from both sides of the above equation: B + C – B = 2×A + B – B which makes C = 2×A
Hint #2
In eq.6, replace A + B with F (from eq.2): E + F = F + C Subtract F from each side of the equation above: E + F – F = F + C – F which makes E = C and which also makes E = C = 2×A
Hint #3
In eq.3, substitute 2×A for E: A + 2×A = B which makes 3×A = B
Hint #4
Substitute 3×A for B in eq.2: F = A + 3×A which makes F = 4×A
Hint #5
Substitute 4×A for F in eq.5: 4×A – D = A + D In the above equation, add D to each side, and subtract A from each side: 4×A – D + D – A = A + D + D – A which means 3×A = 2×D Divide both sides by 2: 3×A ÷ 2 = 2×D ÷ 2 which makes 1½×A = D
Solution
Substitute 3×A for B, 2×A for C and E, 1½×A for D, and 4×A for F in eq.1: A + 3×A + 2×A + 1½×A + 2×A + 4×A = 27 which becomes 13½×A = 27 Divide both sides of the equation above by 13½: 13½×A ÷ 13½ = 27 ÷ 13½ which means A = 2 making B = 3×A = 3 × 2 = 6 C = E = 2×A = 2 × 2 = 4 D = 1½×A = 1½ × 2 = 3 F = 4×A = 4 × 2 = 8 and ABCDEF = 264348