Puzzle for March 27, 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
Add C to both sides of eq.5: D + E – C + C = B + C + C which becomes D + E = B + 2×C In eq.2, replace D + E with B + 2×C: B + F = C + B + 2×C Subtract B from each side of the equation above: B + F – B = C + B + 2×C – B which makes eq.2a) F = 3×C
Hint #2
Add B to both sides of eq.6: A + B + E + B = C – B + B which becomes eq.6a) A + 2×B + E = C In eq.4, replace C with A + 2×B + E (from eq.6a): A + D = A + 2×B + E + E Subtract A from each side of the above equation: A + D – A = A + 2×B + E + E – A which becomes eq.4a) D = 2×B + 2×E
Hint #3
In eq.5, substitute 2×B + 2×E for D (from eq.4a): 2×B + 2×E + E – C = B + C In the equation above, add C to both sides, and subtract B from both sides: 2×B + 2×E + E – C + C – B = B + C + C – B which becomes B + 3×E = 2×C Divide each side by 2: (B + 3×E) ÷ 2 = 2×C ÷ 2 which becomes eq.5a) ½×B + 1½×E = C
Hint #4
Substitute ½×B + 1½×E for C (from eq.5a) in eq.6a: A + 2×B + E = ½×B + 1½×E Subtract 2×B and E from both sides of the above equation: A + 2×B + E – 2×B – E = ½×B + 1½×E – 2×B – E which becomes eq.6b) A = ½×E – 1½×B
Hint #5
Substitute 2×B + 2×E for D (from eq.4a), ½×E – 1½×B for A (from eq.6b), and ½×B + 1½×E for C (from eq.5a) in eq.3: B + 2×B + 2×E = ½×E – 1½×B + ½×B + 1½×E which becomes 3×B + 2×E = –B + 2×E In the equation above, subtract 2×E from both sides, and add B to each side: 3×B + 2×E – 2×E + B = –B + 2×E – 2×E + B which simplifies to 4×B = 0 which means B = 0
Hint #6
Substitute 0 for B in eq.6b: A = ½×E – 1½×0 which makes A = ½×E
Hint #7
Substitute 0 for B in eq.5a: ½×0 + 1½×E = C which makes 1½×E = C
Hint #8
Substitute 0 for B in eq.4a: D = 2×0 + 2×E which makes D = 2×E
Hint #9
Substitute (1½×E) for C in eq.2a: F = 3×(1½×E) which makes F = 4½×E
Solution
Substitute ½×E for A, 0 for B, 1½×E for C, 2×E for D, and 4½×E for F in eq.1: ½×E + 0 + 1½×E + 2×E + E + 4½×E = 19 which simplifies to 9½×E = 19 Divide both sides of the above equation by 9½: 9½×E ÷ 9½ = 19 ÷ 9½ which means E = 2 making A = ½×E = ½ × 2 = 1 C = 1½×E = 1½ × 2 = 3 D = 2×E = 2 × 2 = 4 F = 4½×E = 4½ × 2 = 9 and ABCDEF = 103429