Puzzle for September 3, 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
Subtract C from both sides of eq.4: B + E - C = A + C + D - C which becomes eq.4a) B + E - C = A + D Subtract B from both sides of eq.5: E + F - B = A + B + D - B which becomes eq.5a) E + F - B = A + D
Hint #2
In eq.5a, replace A + D with B + E - C (from eq.4a): E + F - B = B + E - C In the above equation, subtract E from both sides, and add B and C to both sides: E + F - B - E + B + C = B + E - C - E + B + C which becomes eq.5b) F + C = 2×B
Hint #3
Add B and D to both sides of eq.3: F - B + B + D = B - D + B + D which becomes eq.3a) F + D = 2×B
Hint #4
In eq.3a, replace 2×B with F + C (from eq.5b): F + D = F + C Subtract F from each side of the equation above: F + D - F = F + C - F which makes D = C
Hint #5
eq.6 may be written as: B = (C + D + E + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × B = 4 × (C + D + E + F) ÷ 4 which becomes 4×B = C + D + E + F which may be written as eq.6a) 2×B + 2×B = C + E + D + F
Hint #6
In eq.6a, substitute F + C for one 2×B (from eq.5b), and F + D for the other 2×B (from eq.3a): F + C + F + D = C + E + D + F Subtract C, F, and D from both sides of the above equation: F + C + F + D - C - F - D = C + E + D + F - C - F - D which simplifies to F = E
Hint #7
Substitute D for C, and F for E in eq.2: A = D + F which may be written as eq.2a) A = F + D
Hint #8
Substitute 2×B for F + D (from eq.3a) into eq.2a: A = 2×B
Hint #9
Substitute 2×B for A in eq.2: eq.2b) 2×B = C + E
Hint #10
eq.1 may be written as: A + B + C + E + F + D = 21 Substitute 2×B for A, 2×B for C + E (from eq.2b), and 2×B for F + D (from eq.3a) into the above equation: 2×B + B + 2×B + 2×B = 21 which makes 7×B = 21 Divide both sides by 7: 7×B ÷ 7 = 21 ÷ 7 which makes B = 3 and makes A = 2×B = 2 × 3 = 6
Hint #11
Substitute 3 for B, 6 for A, and C for D in eq.4a: 3 + E - C = 6 + C In the above equation, subtract 3 from both sides, and add C to both sides: 3 + E - C - 3 + C = 6 + C - 3 + C which becomes eq.4b) E = 3 + 2×C
Hint #12
Substitute 3 for B, and 3 + 2×C for E (from eq.4b) in eq.2b: 2×3 = C + 3 + 2×C which becomes 6 = 3×C + 3 Subtract 3 from each side of the equation above: 6 - 3 = 3×C + 3 - 3 which makes 3 = 3×C Divide both sides by 3: 3 ÷ 3 = 3×C ÷ 3 which makes 1 = C and also makes 1 = C = D
Solution
Substitute 1 for C in eq.4b: E = 3 + 2×1 which becomes E = 3 + 2 which makes E = 5 and also makes F = E = 5 and ABCDEF = 631155