Puzzle for February 5, 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.5, replace C + E with B + D (from eq.3): D + F = B + B + D Subtract D from each side of the above equation: D + F – D = B + B + D – D which becomes eq.5a) F = 2×B
Hint #2
In eq.4, replace F with 2×B: A + 2×B = B + E Subtract B from each side of the equation above: A + 2×B – B = B + E – B which becomes eq.4a) A + B = E
Hint #3
In eq.6, substitute 2×B for F, and substitute A + B for E (from eq.4a): 2×B = A + A + B Subtract B from both sides of the equation above: 2×B – B = A + A + B – B which makes B = 2×A
Hint #4
Substitute (2×A) for B in eq.5a: F = 2×(2×A) which makes F = 4×A
Hint #5
Substitute 2×A for B in eq.4a: A + 2×A = E which makes 3×A = E
Hint #6
Substitute 2×A for B, and 3×A for E in eq.3: 2×A + D = C + 3×A Subtract 2×A from both sides of the above equation: 2×A + D – 2×A = C + 3×A – 2×A which becomes eq.3a) D = C + A
Hint #7
Substitute C + A for D (from eq.3a), 2×A for B, and 4×A for F in eq.2: C + C + A = 2×A + 4×A which becomes 2×C + A = 6×A Subtract A from each side of the above equation: 2×C + A – A = 6×A – A which means 2×C = 5×A Divide both sides by 2: 2×C ÷ 2 = 5×A ÷ 2 which makes C = 2½×A
Hint #8
Substitute 2×A for B, 2½×A for C, and 3×A for E in eq.3: 2×A + D = 2½×A + 3×A which becomes 2×A + D = 5½×A Subtract 2×A from each side of the equation above: 2×A + D – 2×A = 5½×A – 2×A which makes D = 3½×A
Solution
Substitute 2×A for B, 2½×A for C, 3½×A for D, 3×A for E, and 4×A for F in eq.1: A + 2×A + 2½×A + 3½×A + 3×A + 4×A = 32 which simplifies to 16×A = 32 Divide each side of the equation above by 16: 16×A ÷ 16 = 32 ÷ 16 which means A = 2 making B = 2×A = 2 × 2 = 4 C = 2½×A = 2½ × 2 = 5 D = 3½×A = 3½ × 2 = 7 E = 3×A = 3 × 2 = 6 F = 4×A = 4 × 2 = 8 and ABCDEF = 245768