Puzzle for April 26, 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 F to both sides of eq.5: C - A + F = A - B - F + F which becomes eq.5a) C - A + F = A - B Subtract F from both sides of eq.6: D - A - F = A - B + F - F which becomes eq.6a) D - A - F = A - B
Hint #2
In eq.6a, replace A - B with C - A + F (from eq.5a): D - A - F = C - A + F Add A and F to both sides of the above equation: D - A - F + A + F = C - A + F + A + F which becomes eq.6b) D = C + 2×F
Hint #3
Add F to both sides of eq.3: C - F + F = E + F + F which becomes eq.3a) C = E + 2×F Subtract F from each side of eq.2: E - F = D + F - F which becomes eq.2a) E - F = D
Hint #4
In eq.6b, replace D with E - F (from eq.2a), and C with E + 2×F (from eq.3a): E - F = E + 2×F + 2×F which becomes E - F = E + 4×F In the above equation, add F to both sides, and subtract E from both sides: E - F + F - E = E + 4×F + F - E which becomes 0 = 5×F which means 0 = F
Hint #5
Substitute 0 for F in eq.3a: C = E + 2×0 which becomes C = E + 0 which makes C = E
Hint #6
Substitute 0 for F in eq.2a: E - 0 = D which makes E = D
Hint #7
Substitute 0 for F, and E for D in eq.4: B - 0 = E - B which becomes B = E - B Add B to both sides of the above equation: B + B = E - B + B which makes 2×B = E Divide both sides by 2: 2×B ÷ 2 = E ÷ 2 which makes B = ½×E
Hint #8
Substitute E for C, ½×E for B, and 0 for F in eq.5: E - A = A - ½×E - 0 which becomes E - A = A - ½×E Add A and ½×E to both sides of the above equation: E - A + A + ½×E = A - ½×E + A + ½×E which makes 1½×E = 2×A Divide both sides by 2: 1½×E ÷ 2 = 2×A ÷ 2 which makes ¾×E = A
Solution
Substitute ¾×E for A, ½×E for B, E for C and D, and 0 for F in eq.1: ¾×E + ½×E + E + E + E + 0 = 34 which simplifies to 4¼×E = 34 Divide both sides of the above equation by 4¼: 4¼×E ÷ 4¼ = 34 ÷ 4¼ which means E = 8 making A = ¾×E = ¾ × 8 = 6 B = ½×E = ½ × 8 = 4 C = D = E = 8 and ABCDEF = 648880