Puzzle for November 25, 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
Subtract B from each side of eq.2: D + F - B = A + B - B which becomes D + F - B = A In eq.4, replace A with D + F - B: D + F - B - F = E + F which becomes eq.4a) D - B = E + F
Hint #2
In eq.3, replace E + F with D - B (from eq.4a): D - B = B + D In the above equation, add B to both sides, and subtract D from both sides: D - B + B - D = B + D + B - D which makes 0 = 2×B which means 0 = B
Hint #3
Add F to both sides of eq.4: A - F + F = E + F + F which becomes eq.4b) A = E + 2×F eq.6 may be written as: E = (A + B + C) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (A + B + C) ÷ 3 which becomes eq.6a) 3×E = A + B + C
Hint #4
In eq.6a, substitute E + 2×F for A (from eq.4b), and 0 for B: 3×E = E + 2×F + 0 + C which becomes 3×E = E + 2×F + C Subtract E and 2×F from each side of the above equation: 3×E - E - 2×F = E + 2×F + C - E - 2×F which becomes eq.6b) 2×E - 2×F = C
Hint #5
In eq.4a, substitute 0 for B: D - 0 = E + F which becomes eq.4c) D = E + F
Hint #6
Substitute (E + F) for D (from eq.4c), 0 for B, and 2×E - 2×F for C (from eq.6b) in eq.5: (E + F) - E = 0 + 2×E - 2×F - (E + F) which becomes F = 2×E - 2×F - E - F which becomes F = E - 3×F Add 3×F to both sides of the equation above: F + 3×F = E - 3×F + 3×F which makes 4×F = E
Hint #7
Substitute 4×F for E in eq.4b: A = 4×F + 2×F which makes A = 6×F
Hint #8
Substitute (4×F) for E in eq.6b: 2×(4×F) - 2×F = C which becomes 8×F - 2×F = C which makes 6×F = C
Hint #9
Substitute 4×F for E in eq.4c: D = 4×F + F which makes D = 5×F
Solution
Substitute 6×F for A and C, 0 for B, 5×F for D, and 4×F for E in eq.1: 6×F + 0 + 6×F + 5×F + 4×F + F = 22 which simplifies to 22×F = 22 Divide both sides of the above equation by 22: 22×F ÷ 22 = 22 ÷ 22 which means F = 1 making A = C = 6×F = 6 × 1 = 6 D = 5×F = 5 × 1 = 5 E = 4×F = 4 × 1 = 4 and ABCDEF = 606541