Puzzle for July 19, 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 B to both sides of eq.3: A - E + E + B = D - B + E + B which becomes eq.3a) A + B = D + E
Hint #2
Subtract the left and right sides of eq.3a from the left and right sides of eq.2, respectively: B + E - (A + B) = A + D - (D + E) which is equivalent to B + E - A - B = A + D - D - E which becomes E - A = A - E Add A and E to both sides of the above equation: E - A + A + E = A - E + A + E which makes 2×E = 2×A Divide both sides by 2: 2×E ÷ 2 = 2×A ÷ 2 which makes E = A
Hint #3
In eq.2, replace E with A: B + A = A + D Subtract A from each side of the equation above: B + A - A = A + D - A which makes B = D
Hint #4
In eq.5, substitute B for D: B + F - C = A + B + C In the above equation, subtract B from both sides, and add C to both sides: B + F - C - B + C = A + B + C - B + C which becomes eq.5a) F = A + 2×C
Hint #5
Add C to both sides of eq.4: F - C + C = B + C + C which becomes eq.4a) F = B + 2×C
Hint #6
Substitute B + 2×C for F (from eq.4a) into eq.5a: B + 2×C = A + 2×C Subtract 2×C from both sides of the equation above: B + 2×C - 2×C = A + 2×C - 2×C which makes B = A and also makes D = B = A = E
Hint #7
Substitute D for E in eq.6: C = D ÷ D which makes C = 1
Hint #8
Substitute 1 for C in eq.5a: F = A + 2×1 which becomes eq.5b) F = A + 2
Hint #9
Substitute A for B and D and E, 1 for C, and A + 2 for F (from eq.5b) in eq.1: A + A + 1 + A + A + A + 2 = 38 which simplifies to 5×A + 3 = 38 Subtract 3 from each side of the above equation: 5×A + 3 - 3 = 38 - 3 which makes 5×A = 35 Divide both sides by 5: 5×A ÷ 5 = 35 ÷ 5 which means A = 7
Solution
Since A = 7, then: B = D = E = A = 7 F = A + 2 = 7 + 2 = 9 (from eq.5b) and ABCDEF = 771779