Puzzle for July 1, 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
Add E and C to both sides of eq.2: D - E + E + C = B - C + E + C which becomes D + C = B + E which may be written as eq.2a) C + D = B + E
Hint #2
In eq.6, replace C + D with B + E (from eq.2a): B + B + E = A + E + F which becomes 2×B + E = A + E + F Subtract E from each side of the equation above: 2×B + E - E = A + E + F - E which becomes eq.6a) 2×B = A + F
Hint #3
Add B and E to both sides of eq.4: A - B + B + E = B - E + B + E which becomes eq.4a) A + E = 2×B
Hint #4
In eq.6a, replace 2×B with A + E (from eq.4a): A + E = A + F which makes E = F
Hint #5
Add D and F to both sides of eq.5: E - D + D + F = B + D - F + D + F which becomes eq.5a) E + F = B + 2×D
Hint #6
In eq.6, substitute B + 2×D for E + F (from eq.5a): B + C + D = A + B + 2×D Subtract B and D from both sides of the equation above: B + C + D - B - D = A + B + 2×D - B - D which simplifies to eq.6b) C = A + D
Hint #7
Add A and E to both sides of eq.3: C - A + A + E = A - E + A + E which becomes eq.3a) C + E = 2×A
Hint #8
Substitute A + D for C (from eq.6b) into eq.3a: A + D + E = 2×A Subtract A from each side of the above equation: A + D + E - A = 2×A - A which becomes eq.3b) D + E = A
Hint #9
Substitute D + E for A (from eq.3b) in eq.6b: C = D + E + D which becomes eq.6c) C = 2×D + E
Hint #10
Substitute 2×D + E for C (from eq.6c) in eq.2a: 2×D + E + D = B + E which becomes 3×D + E = B + E Subtract E from each side of the equation above: 3×D + E - E = B + E - E which makes 3×D = B
Hint #11
Substitute E for F, and 3×D for B in eq.5a: E + E = 3×D + 2×D which makes 2×E = 5×D Divide both sides of the above equation by 2: 2×E ÷ 2 = 5×D ÷ 2 which makes E = 2½×D and also makes E = F = 2½×D
Hint #12
Substitute 2½×D for E in eq.6c: C = 2×D + 2½×D which makes C = 4½×D
Hint #13
Substitute 2½×D for E in eq.3b: D + 2½×D = A which makes 3½×D = A
Solution
Substitute 3½×D for A, 3×D for B, 4½×D for C, and 2½×D for E and F in eq.1: 3½×D + 3×D + 4½×D + D + 2½×D + 2½×D = 34 which simplifies to 17×D = 34 Divide both sides of the above equation by 17: 17×D ÷ 17 = 34 ÷ 17 which means D = 2 making A = 3½×D = 3½ × 2 = 7 B = 3×D = 3 × 2 = 6 C = 4½×D = 4½ × 2 = 9 E = F = 2½×D = 2½ × 2 = 5 and ABCDEF = 769255