Puzzle for July 14, 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.6 may be written as: A - B = (C + D) ÷ 2 Multiply both sides of the above equation by 2: 2 × (A - B) = 2 × (C + D) ÷ 2 which becomes eq.6a) 2×A - 2×B = C + D
Hint #2
Subtract C from both sides of eq.3: E - C - C = C + D - C which becomes eq.3a) E - 2×C = D
Hint #3
In eq.2, replace D with E - 2×C (from eq.3a): E - 2×C = B + C Subtract C from each side of the equation above: E - 2×C - C = B + C - C which becomes eq.2a) E - 3×C = B
Hint #4
In eq.4, substitute E - 3×C for B (from eq.2a), and E - 2×C for D (from eq.3a): A + C = E - 3×C + E - 2×C + E - A which becomes A + C = 3×E - 5×C - A In the above equation, add A to both sides, and subtract C from both sides: A + C + A - C = 3×E - 5×C - A + A - C which becomes 2×A = 3×E - 6×C Divide both sides by 2: 2×A ÷ 2 = (3×E - 6×C) ÷ 2 which becomes eq.4a) A = 1½×E - 3×C
Hint #5
In eq.6a, substitute (1½×E - 3×C) for A (from eq.4a), (E - 3×C) for B (from eq.2a), and E - 2×C for D (from eq.3a): 2×(1½×E - 3×C) - 2×(E - 3×C) = C + E - 2×C which becomes 3×E - 6×C - 2×E + 6×C = -C + E which becomes E = -C + E Subtract E from each side of the equation above: E - E = -C + E - E which makes 0 = -C which means 0 = C
Hint #6
Substitute 0 for C in eq.4a: A = 1½×E - 3×0 which becomes A = 1½×E - 0 which makes A = 1½×E
Hint #7
Substitute 0 for C in eq.2a: E - 3×0 = B which becomes E - 0 = B which makes E = B
Hint #8
Substitute 0 for C in eq.3a: E - 2×0 = D which becomes E - 0 = D which makes E = D
Hint #9
eq.5 may be written as: F = (A + B + C) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × (A + B + C) ÷ 3 which becomes eq.5a) 3×F = A + B + C
Hint #10
In eq.5a, substitute 1½×E for A, E for B, and 0 for C: 3×F = 1½×E + E + 0 which becomes 3×F = 2½×E Divide both sides of the above equation by 3: 3×F ÷ 3 = 2½×E ÷ 3 which becomes eq.5b) F = ⅚×E
Solution
Substitute 1½×E for A, E for B and D, 0 for C, and ⅚×E for F in eq.1: 1½×E + E + 0 + E + E + ⅚×E = 32 which simplifies to 5⅓×E = 32 Divide both sides of the above equation by 5⅓: 5⅓×E ÷ 5⅓ = 32 ÷ 5⅓ which means E = 6 making A = 1½×E = 1½ × 6 = 9 B = D = E = 6 F = ⅚×E = ⅚ × 6 = 5 and ABCDEF = 960665