Puzzle for February 2, 2023 ( )
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.
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Hint #1
In eq.1, replace A + B + C with D + E + F (from eq.6): D + E + F + D + E + F = 32 which is equivalent to 2×(D + E + F) = 32 Divide both sides of the above equation by 2: 2×(D + E + F) ÷ 2 = 32 ÷ 2 which means eq.1a) D + E + F = 16
Hint #2
In eq.1a, replace D + E with F (from eq.2): F + F = 16 which makes 2×F = 16 Divide both sides of the above equation by 2: 2×F ÷ 2 = 16 ÷ 2 which makes F = 8
Hint #3
In eq.4, substitute B + D for C (from eq.3): B + B + D = A + D which becomes 2×B + D = A + D Subtract D from each side of the equation above: 2×B + D - D = A + D - D which makes eq.4a) 2×B = A
Hint #4
Substitute 2×B for A, and 8 for F in eq.5: 2×B + B = C + 8 - 2×B which becomes 3×B = C + 8 - 2×B In the equation above, add 2×B to both sides, and subtract 8 from both sides: 3×B + 2×B - 8 = C + 8 - 2×B + 2×B - 8 which becomes eq.5a) 5×B - 8 = C
Hint #5
Substitute 16 for D + E + F (from eq.1a) in eq.6: eq.6a) 16 = A + B + C
Hint #6
Substitute 2×B for A, and 5×B - 8 for C (from eq.5a) in eq.6a: 16 = 2×B + B + 5×B - 8 which becomes 16 = 8×B - 8 Add 8 to both sides of the above equation: 16 + 8 = 8×B - 8 + 8 which makes 24 = 8×B Divide both sides by 8: 24 ÷ 8 = 8×B ÷ 8 which makes 3 = B
Hint #7
Substitute 3 for B in eq.4a: 2×3 = A which makes 6 = A
Hint #8
Substitute 3 for B in eq.5a: 5×3 - 8 = C which becomes 15 - 8 = C which makes 7 = C
Hint #9
Substitute 7 for C, and 3 for B in eq.3: 7 = 3 + D Subtract 3 from each side of the above equation: 7 - 3 = 3 + D - 3 which makes 4 = D
Solution
Substitute 8 for F, and 4 for D in eq.2: 8 = 4 + E Subtract 4 from each side of the above equation: 8 - 4 = 4 + E - 4 which makes 4 = E and makes ABCDEF = 637448