Puzzle for October 18, 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
eq.5 may be written as: B + C = A + F + D + E In the above equation, replace A + F with D (from eq.2): B + C = D + D + E which becomes eq.5a) B + C = 2×D + E
Hint #2
In eq.6, replace B + C with 2×D + E (from eq.5a): D + E - C = 2×D + E - D which becomes D + E - C = D + E Subtract D and E from both sides of the above equation: D + E - C - D - E = D + E - D - E which makes -C = 0 which means C = 0
Hint #3
In eq.3, substitute 0 for C: E + F = 0 + D which becomes eq.3a) E + F = D
Hint #4
Substitute E + F for D (from eq.3a) into eq.2: E + F = A + F Subtract F from each side of the above equation: E + F - F = A + F - F which makes E = A
Hint #5
Substitute (A + F) for D (from eq.2), and 0 for C in eq.4: A - (A + F) = 0 + F - A which becomes A - A - F = F - A which becomes -F = F - A Add F and A to both sides of the equation above: -F + F + A = F - A + F + A which makes A = 2×F and also makes E = A = 2×F
Hint #6
Substitute 2×F for A in eq.2: D = 2×F + F which makes D = 3×F
Hint #7
Substitute 0 for C, 3×F for D, and 2×F for E in eq.5a: B + 0 = 2×(3×F) + 2×F which becomes B = 6×F + 2×F which makes B = 8×F
Solution
Substitute 2×F for A and E, 8×F for B, 0 for C, and 3×F for D in eq.1: 2×F + 8×F + 0 + 3×F + 2×F + F = 16 which simplifies to 16×F = 16 Divide both sides of the above equation by 16: 16×F ÷ 16 = 16 ÷ 16 which means F = 1 making A = E = 2×F = 2 × 1 = 2 B = 8×F = 8 × 1 = 8 D = 3×F = 3 × 1 = 3 and ABCDEF = 280321