Puzzle for December 5, 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 D to both sides of eq.3: C + D + D = B - D + D which becomes eq.3a) C + 2×D = B
Hint #2
In eq.6, replace B with C + 2×D (from eq.3a): A + D = C + 2×D + C which becomes A + D = 2×C + 2×D Subtract D from each side of the equation above: A + D - D = 2×C + 2×D - D which becomes eq.6a) A = 2×C + D
Hint #3
In eq.2, replace A with 2×C + D (from eq.6a): 2×C + D = D + E Subtract D from both sides of the equation above: 2×C + D - D = D + E - D which makes 2×C = E
Hint #4
In eq.5, substitute 2×C for E: B - 2×C = C - B Add 2×C and B to both sides of the above equation: B - 2×C + 2×C + B = C - B + 2×C + B which makes 2×B = 3×C Divide both sides by 2: 2×B ÷ 2 = 3×C ÷ 2 which makes B = 1½×C
Hint #5
Substitute 1½×C for B in eq.3a: C + 2×D = 1½×C Subtract C from each side of the above equation: C + 2×D - C = 1½×C - C which makes 2×D = ½×C Divide both sides by 2: 2×D ÷ 2 = ½×C ÷ 2 which makes D = ¼×C
Hint #6
Substitute ¼×C for D, and 2×C for E in eq.2: A = ¼×C + 2×C which makes A = 2¼×C
Hint #7
Substitute 2×C for E, and 2¼×C for A in eq.4: F - 2×C = 2×C - 2¼×C which becomes F - 2×C = -¼×C Add 2×C to both sides of the equation above: F - 2×C + 2×C = -¼×C + 2×C which makes F = 1¾×C
Solution
Substitute 2¼×C for A, 1½×C for B, ¼×C for D, 2×C for E, and 1¾×C for F in eq.1: 2¼×C + 1½×C + C + ¼×C + 2×C + 1¾×C = 35 which simplifies to 8¾×C = 35 Divide both sides of the above equation by 8¾: 8¾×C ÷ 8¾ = 35 ÷ 8¾ which means C = 4 making A = 2¼×C = 2¼ × 4 = 9 B = 1½×C = 1½ × 4 = 6 D = ¼×C = ¼ × 4 = 1 E = 2×C = 2 × 4 = 8 F = 1¾×C = 1¾ × 4 = 7 and ABCDEF = 964187