Puzzle for July 21, 2019 ( )
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.2, replace F with A + B + E (from eq.4): D + A + B + E = A + B + C Subtract both A and B from each side of the above equation: D + A + B + E - A - B = A + B + C - A - B which simplifies to eq.4a) D + E = C
Hint #2
In eq.6, replace F with A + B + E (from eq.4): A + C + D = E + A + B + E which becomes A + C + D = A + B + 2×E Subtract A from both sides of the above equation: A + C + D - A = A + B + 2×E - A which becomes eq.6a) C + D = B + 2×E
Hint #3
In eq.6a, replace C with D + E (from eq.4a): D + E + D = B + 2×E which becomes 2×D + E = B + 2×E Express E in terms of B and D by subtracting B and E from both sides of the above equation: 2×D + E - B - E = B + 2×E - B - E which becomes eq.6b) 2×D - B = E
Hint #4
In eq.6a, substitute (2×D - B) for E (from eq.6b): C + D = B + 2×(2×D - B) which is the same as C + D = B + 4×D - 2×B which becomes C + D = 4×D - B Express C in terms of B and D by subtracting D from both sides of the above equation: C + D - D = 4×D - B - D which becomes eq.6c) C = 3×D - B
Hint #5
Add C to both sides of eq.3: C - D + E + C = B - C + C which becomes 2×C - D + E = B Substitute (3×D - B) for C (from eq.6c), and 2×D - B for E (from eq.6b) in the above equation: 2×(3×D - B) - D + 2×D - B = B which becomes 6×D - 2×B + D - B = B which becomes 7×D - 3×B = B Add 3×B to both sides: 7×D - 3×B + 3×B = B + 3×B which means 7×D = 4×B Divide both sides by 4: 7×D ÷ 4 = 4×B ÷ 4 which makes 1¾×D = B
Hint #6
Substitute 1¾×D for B in eq.6c: C = 3×D - 1¾×D which makes C = 1¼×D
Hint #7
Substitute 1¾×D for B in eq.6b: 2×D - 1¾×D = E which makes ¼×D = E
Hint #8
Substitute 1¾×D for B, 1¼×D for C, and ¼×D for E in eq.5: 1¾×D - A = 1¼×D + ¼×D which becomes 1¾×D - A = 1½×D Add A to each side, and subtract 1½×D from each side: 1¾×D - A + A - 1½×D = 1½×D + A - 1½×D which makes ¼×D = A
Hint #9
Substitute ¼×D for A and E, 1¾×D for B, and ¼×D for E in eq.4: F = ¼×D + 1¾×D + ¼×D which makes F = 2¼×D
Solution
Substitute ¼×D for A for E, 1¾×D for B, 1¼×D for C, ¼×D for E, and 2¼×D for F in eq.1: ¼×D + 1¾×D + 1¼×D + D + ¼×D + 2¼×D = 27 which simplifies to 6¾×D = 27 Divide both sides by 6¾: 6¾×D ÷ 6¾ = 27 ÷ 6¾ which means D = 4 making A = E = ¼×D = ¼ × 4 = 1 B = 1¾×D = 1¾ × 4 = 7 C = 1¼×D = 1¼ × 4 = 5 F = 2¼×D = 2¼ × 4 = 9 and ABCDEF = 175419