Puzzle for July 27, 2021 ( )
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 A + B with D (from eq.2): eq.4a) E = D + F
Hint #2
In eq.3, replace D + F with E (from eq.4a): C = E
Hint #3
In eq.5, replace D with A + B (from eq.2): B + F = A + A + B which becomes B + F = 2×A + B Subtract B from each side of the above equation: B + F – B = 2×A + B – B which makes F = 2×A
Hint #4
In eq.6, substitute 2×A for F, and A + B for D (from eq.2): C + 2×A – B = A + B + A + B which becomes C + 2×A – B = 2×A + 2×B In the above equation, subtract 2×A from both sides, and add B to both sides: C + 2×A – B – 2×A + B = 2×A + 2×B – 2×A + B which makes C = 3×B and also makes eq.6a) E = C = 3×B
Hint #5
In eq.3, substitute 3×B for C (from eq.6a), A + B for D (from eq.2), and 2×A for F: 3×B = A + B + 2×A which becomes 3×B = B + 3×A Subtract B from both sides of the equation above: 3×B – B = B + 3×A – B which makes 2×B = 3×A Divide both sides by 2: 2×B ÷ 2 = 3×A ÷ 2 which makes B = 1½×A
Hint #6
Substitute (1½×A) for B in eq.6a: E = C = 3×(1½×A) which makes E = C = 4½×A
Hint #7
Substitute 4½×A for C, and 2×A for F in eq.3: 4½×A = D + 2×A Subtract 2×A from each side of the equation above: 4½×A – 2×A = D + 2×A – 2×A which makes 2½×A = D
Solution
Substitute 1½×A for B, 4½×A for C and E, 2½×A for D, and 2×A for F in eq.1: A + 1½×A + 4½×A + 2½×A + 4½×A + 2×A = 32 which simplifies to 16×A = 32 Divide both sides of the above equation by 16: 16×A ÷ 16 = 32 ÷ 16 which means A = 2 making B = 1½×A = 1½ × 2 = 3 C = E = 4½×A = 4½ × 2 = 9 D = 2½×A = 2½ × 2 = 5 F = 2×A = 2 × 2 = 4 and ABCDEF = 239594