Puzzle for July 22, 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
eq.4 may be written as: B + A + E = C + F In the equation above, replace A + E with B + C (from eq.2): B + B + C = C + F Subtract C from both sides: B + B + C - C = C + F - C which makes 2×B = F
Hint #2
In eq.6, replace F with 2×B: D + 2×B = A + B Subtract B from each side of the above equation: D + 2×B - B = A + B - B which becomes D + B = A which may be written as eq.6a) B + D = A In eq.5, replace B + D with A (from eq.6a): E = A
Hint #3
Substitute B + D for E (from eq.5), and 2×B for F in eq.3: C + D = B + B + D + 2×B which becomes C + D = 4×B + D Subtract D from both sides of the above equation: C + D - D = 4×B + D - D which simplifies to C = 4×B
Hint #4
Substitute 4×B for C, and A for E in eq.2: B + 4×B = A + A which makes 5×B = 2×A Divide both sides of the above equation by 2: 5×B ÷ 2 = 2×A ÷ 2 which makes 2½×B = A and which also makes E = A = 2½×B
Hint #5
Substitute 2½×B for E in eq.5: 2½×B = B + D Subtract B from both sides: 2½×B - B = B + D - B which makes 1½×B = D
Solution
Substitute 2½×B for A and E, 4×B for C, 1½×B for D, and 2×B for F in eq.1: 2½×B + B + 4×B + 1½×B + 2½×B + 2×B = 27 which becomes 13½×B = 27 Divide both sides of the above equation by 13½: 13½×B ÷ 13½ = 27 ÷ 13½ which means B = 2 making A = E = 2½×B = 2½ × 2 = 5 C = 4×B = 4 × 2 = 8 D = 1½×B = 1½ × 2 = 3 F = 2×B = 2 × 2 = 4 and ABCDEF = 528354