Puzzle for December 18, 2022 ( )
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
eq.6 may be written as: C = (A + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × C = 3 × (A + E + F) ÷ 3 which becomes eq.6a) 3×C = A + E + F
Hint #2
In eq.5, replace D + F with B + C - F (from eq.3): B + E = C + B + C - F - B which becomes B + E = 2×C - F In the above equation, subtract B from both sides, and add F to both sides: B + E - B + F = 2×C - F - B + F which becomes eq.5a) E + F = 2×C - B
Hint #3
In eq.6a, replace E + F with 2×C - B (from eq.5a): 3×C = A + 2×C - B Subtract 2×C from each side of the equation above: 3×C - 2×C = A + 2×C - B - 2×C which becomes eq.6b) C = A - B
Hint #4
In eq.2, substitute A - B for C (from eq.6b): E + F = A + B + A - B which becomes eq.2a) E + F = 2×A
Hint #5
Substitute 2×A for E + F (from eq.2a) in eq.6a: 3×C = A + 2×A which makes 3×C = 3×A Divide both sides of the above equation by 3: 3×C ÷ 3 = 3×A ÷ 3 which makes C = A
Hint #6
Substitute A for C in eq.6b: A = A - B Subtract A from each side of the equation above: A - A = A - B - A which makes 0 = -B which means 0 = B
Hint #7
Substitute 0 for B, and A for C in eq.3: D + F = 0 + A - F which becomes D + F = A - F Add F to both sides of the equation above: D + F + F = A - F + F which becomes eq.3a) D + 2×F = A
Hint #8
Substitute 0 for B in eq.4: A - D = 0 + D - F which becomes A - D = D - F Add D to both sides of the above equation: A - D + D = D - F + D which becomes eq.4a) A = 2×D - F
Hint #9
Substitute 2×D - F for A from eq.4a into eq.3a: D + 2×F = 2×D - F In the equation above, subtract D from both sides, and add F to both sides: D + 2×F - D + F = 2×D - F - D + F which makes 3×F = D
Hint #10
Substitute 3×F for D in eq.3a: 3×F + 2×F = A which makes 5×F = A and also makes 5×F = A = C
Hint #11
Substitute (5×F) for A in eq.2a: E + F = 2×(5×F) which makes E + F = 10×F Subtract F from both sides of the equation above: E + F - F = 10×F - F which makes E = 9×F
Solution
Substitute 5×F for A and C, 0 for B, 3×F for D, and 9×F for E in eq.1: 5×F + 0 + 5×F + 3×F + 9×F + F = 23 which simplifies to 23×F = 23 Divide both sides of the above equation by 23: 23×F ÷ 23 = 23 ÷ 23 which means F = 1 making A = C = 5×F = 5 × 1 = 5 D = 3×F = 3 × 1 = 3 E = 9×F = 9 × 1 = 9 and ABCDEF = 505391