Puzzle for September 6, 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 and E to both sides of eq.3: D - E + D + E = C - D + D + E which becomes eq.3a) 2×D = C + E
Hint #2
In eq.4, replace C + E with 2×D (from eq.3a): 2×D = D + F Subtract D from each side of the equation above: 2×D - D = D + F - D which makes D = F
Hint #3
In eq.5, replace A with D + E (from eq.2), and F with D: D + E + E = B + D which becomes D + 2×E = B + D Subtract D from each side of the equation above: D + 2×E - D = B + D - D which makes 2×E = B
Hint #4
In eq.6, substitute 2×E for B: 2×E - D = C - 2×E In the above equation, add D and 2×E to both sides, and subtract C from both sides: 2×E - D + D + 2×E - C = C - 2×E + D + 2×E - C which becomes eq.6a) 4×E - C = D
Hint #5
Substitute (4×E - C) for D (from eq.6a) in eq.3a: 2×(4×E - C) = C + E which becomes 8×E - 2×C = C + E In the above equation, add 2×C to both sides, and subtract E from both sides: 8×E - 2×C + 2×C - E = C + E + 2×C - E which makes 7×E = 3×C Divide both sides by 3: 7×E ÷ 3 = 3×C ÷ 3 which makes 2⅓×E = C
Hint #6
Substitute 2⅓×E for C in eq.6a: 4×E - 2⅓×E = D which makes 1⅔×E = D and also makes 1⅔×E = D = F
Hint #7
Substitute 1⅔×E for D in eq.2: A = 1⅔×E + E which makes A = 2⅔×E
Solution
Substitute 2⅔×E for A, 2×E for B, 2⅓×E for C, 1⅔×E for D and F in eq.1: 2⅔×E + 2×E + 2⅓×E + 1⅔×E + E + 1⅔×E = 34 which simplifies to 11⅓×E = 34 Divide both sides of the above equation by 11⅓: 11⅓×E ÷ 11⅓ = 34 ÷ 11⅓ which means E = 3 making A = 2⅔×E = 2⅔ × 3 = 8 B = 2×E = 2 × 3 = 6 C = 2⅓×E = 2⅓ × 3 = 7 D = F = 1⅔×E = 1⅔ × 3 = 5 and ABCDEF = 867535