Puzzle for March 2, 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
In eq.2, replace A + B with D + E (from eq.1): E + F = D + E - E which becomes eq.2a) E + F = D
Hint #2
eq.4 may be written as: D = (C + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × D = 3 × (C + E + F) ÷ 3 which becomes eq.4a) 3×D = C + E + F
Hint #3
In eq.4a, replace C + E with D - E + F (from eq.3): 3×D = D - E + F + F which becomes 3×D = D - E + 2×F Subtract D from each side of the equation above: 3×D - D = D - E + 2×F - D which becomes eq.4b) 2×D = -E + 2×F
Hint #4
In eq.4b, substitute (E + F) for D (from eq.2a): 2×(E + F) = -E + 2×F which becomes 2×E + 2×F = -E + 2×F In the above equation, subtract 2×F from both sides, and add E to both sides: 2×E + 2×F - 2×F + E = -E + 2×F - 2×F + E which becomes 3×E = 0 which makes E = 0
Hint #5
Substitute 0 for E in eq.2a: 0 + F = D which makes F = D
Hint #6
Substitute 0 for E, and D for F in eq.3: C + 0 = D - 0 + D which makes C = 2×D
Hint #7
Substitute 2×D for C, and D for F in eq.6: A = 2×D ÷ (D + D) which becomes A = 2×D ÷ (2×D) which makes A = 1
Hint #8
Substitute 0 for E, 1 for A, and 2×D for C in eq.5: B + 0 - 1 = 2×D ÷ D which becomes B - 1 = 2 Add 1 to both sides of the above equation: B - 1 + 1 = 2 + 1 which makes B = 3
Solution
Substitute 1 for A, 3 for B, and 0 for E in eq.1: 1 + 3 = D + 0 which makes 4 = D and also makes 4 = D = F making C = 2×D = 2 × 4 = 8 and ABCDEF = 138404