Puzzle for January 13, 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.
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Hint #1
Add A and E to both sides of eq.4: C + E + A + E = B - A - E + A + E which becomes eq.4a) C + 2×E + A = B
Hint #2
In eq.5, replace B with C + 2×E + A (from eq.4a): C + 2×E + A + F = A + C + D + E Subtract C, E, and A from both sides of the above equation: C + 2×E + A + F - C - E - A = A + C + D + E - C - E - A which simplifies to eq.5a) E + F = D
Hint #3
In eq.2, replace D with E + F (from eq.5a): E + F = A + F Subtract F from each side of the equation above: E + F - F = A + F - F which makes A = E
Hint #4
In eq.4a, substitute A for E: C + 2×A + A = B which becomes eq.4b) C + 3×A = B
Hint #5
Substitute (C + 3×A) for B (from eq.4b) in eq.3: (C + 3×A) + C = D - (3×A + C) which becomes 2×C + 3×A = D - 3×A - C Add 3×A and C to both sides of the above equation: 3×A + 2×C + 3×A + C = D - 3×A - C + 3×A + C which becomes eq.3a) 6×A + 3×C = D
Hint #6
Substitute A + F for D (from eq.2) into eq.3a: 6×A + 3×C = A + F Subtract A from both sides of the equation above: 6×A + 3×C - A = A + F - A which becomes eq.3b) 5×A + 3×C = F
Hint #7
Substitute 5×A + 3×C for F (from eq.3b), C + 3×A for B (from eq.4b), and A for E in eq.6: 5×A + 3×C - A - C = C + 3×A + A + A which becomes 4×A + 2×C = C + 5×A Subtract C and 4×A from both sides of the equation above: 4×A + 2×C - C - 4×A = C + 5×A - C - 4×A which makes C = A
Hint #8
Substitute A for C in eq.4b: A + 3×A = B which makes 4×A = B
Hint #9
Substitute A for C in eq.3a: 6×A + 3×A = D which makes 9×A = D
Hint #10
Substitute A for C in eq.3b: 5×A + 3×A = F which makes 8×A = F
Solution
Substitute 4×A for B, A for C and E, 9×A for D, and 8×A for F in eq.1: A + 4×A + A + 9×A + A + 8×A = 24 which simplifies to 24×A = 24 Divide both sides of the above equation by 24: 24×A ÷ 24 = 24 ÷ 24 which means A = 1 making B = 4×A = 4 × 1 = 4 C = E = A = 1 D = 9×A = 9 × 1 = 9 F = 8×A = 8 × 1 = 8 and ABCDEF = 141918