Puzzle for January 27, 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.
Scratchpad
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Hint #1
Add A to both sides of eq.3: F - A + A = A + C + A which makes eq.3a) F = 2×A + C Add B to both sides of eq.6: D + F - B + B = A + B + E + B which becomes eq.6a) D + F = A + 2×B + E
Hint #2
In eq.6a, replace F with 2×A + C (from eq.3a): D + 2×A + C = A + 2×B + E Subtract A from each side of the equation above: D + 2×A + C - A = A + 2×B + E - A which becomes eq.6b) D + A + C = 2×B + E
Hint #3
Add E to both sides of eq.5: B + E + E = A + C + D - E + E which becomes B + 2×E = A + C + D which may be written as eq.5a) B + 2×E = D + A + C
Hint #4
In eq.5a, replace D + A + C with 2×B + E (from eq.6b): B + 2×E = 2×B + E Subtract B and E from each side of the above equation: B + 2×E - B - E = 2×B + E - B - E which makes E = B
Hint #5
In eq.4, substitute B + C for D (from eq.2), and B for E: C + B + C = B + F which becomes 2×C + B = B + F Subtract B from both sides of the above equation: 2×C + B - B = B + F - B which makes eq.4a) 2×C = F
Hint #6
Substitute 2×C for F in eq.3a: 2×C = 2×A + C Subtract C from both sides of the equation above: 2×C - C = 2×A + C - C which makes C = 2×A
Hint #7
Substitute (2×A) for C in eq.4a: 2×(2×A) = F which makes 4×A = F
Hint #8
Substitute 4×A for F, and B for E in eq.6a: D + 4×A = A + 2×B + B which becomes D + 4×A = A + 3×B Subtract 4×A from each side of the equation above: D + 4×A - 4×A = A + 3×B - 4×A which becomes eq.6c) D = 3×B - 3×A
Hint #9
Substitute 3×B - 3×A for D (from eq.6c), and 2×A for C into eq.2: 3×B - 3×A = B + 2×A In the above equation, add 3×A to both sides, and subtract B from both sides: 3×B - 3×A + 3×A - B = B + 2×A + 3×A - B which becomes 2×B = 5×A Divide both sides by 2: 2×B ÷ 2 = 5×A ÷ 2 which makes B = 2½×A and also makes E = B = 2½×A
Hint #10
Substitute 2½×A for B, and 2×A for C in eq.2: D = 2½×A + 2×A which makes D = 4½×A
Solution
Substitute 2½×A for B and E, 2×A for C, 4½×A for D, and 4×A for F in eq.1: A + 2½×A + 2×A + 4½×A + 2½×A + 4×A = 33 which simplifies to 16½×A = 33 Divide both sides of the above equation by 16½: 16½×A ÷ 16½ = 33 ÷ 16½ which means A = 2 making B = E = 2½×A = 2½ × 2 = 5 C = 2×A = 2 × 2 = 4 D = 4½×A = 4½ × 2 = 9 F = 4×A = 4 × 2 = 8 and ABCDEF = 254958