Puzzle for November 16, 2022 ( )
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 to both sides of eq.4: B + E + E = A - E + E which becomes eq.4a) B + 2×E = A Add B and E to both sides of eq.3: A - B + B + E = F - E + B + E which becomes eq.3a) A + E = F + B
Hint #2
In eq.3a, replace A with B + 2×E (from eq.4a): B + 2×E + E = F + B which becomes B + 3×E = F + B Subtract B from each side of the above equation: B + 3×E - B = F + B - B which makes 3×E = F
Hint #3
In eq.2, replace B + F with A + E (from eq.3a): C + E = A + E Subtract E from each side of the equation above: C + E - E = A + E - E which makes C = A
Hint #4
In eq.6, substitute 3×E for F, and A for C: 3×E - B = A ÷ A which becomes 3×E - B = 1 In the above equation, add B to both sides, and subtract 1 from both sides: 3×E - B + B - 1 = 1 + B - 1 which becomes eq.6a) 3×E - 1 = B
Hint #5
Substitute 3×E - 1 for B (from eq.6a) in eq.4a: 3×E - 1 + 2×E = A which makes 5×E - 1 = A and also makes eq.4b) 5×E - 1 = A = C
Hint #6
Substitute 3×E for F, and 3×E - 1 for B (from eq.6a) in eq.5: E + 3×E = 3×E - 1 + D - 3×E which becomes 4×E = -1 + D Add 1 to both sides of the above equation: 4×E + 1 = -1 + D + 1 which becomes eq.5a) 4×E + 1 = D
Hint #7
Substitute 5×E - 1 for A and C (from eq.4b), 3×E - 1 for B (from eq.6a), 4×E + 1 for D (from eq.5a), and 3×E for F in eq.1: 5×E - 1 + 3×E - 1 + 5×E - 1 + 4×E + 1 + E + 3×E = 40 which simplifies to 21×E - 2 = 40 Add 2 to both sides of the above equation: 21×E - 2 + 2 = 40 + 2 which makes 21×E = 42 Divide both sides by 21: 21×E ÷ 21 = 42 ÷ 21 which means E = 2
Solution
Since E = 2, then: A = C = 5×E - 1 = 5×2 - 1 = 10 - 1 = 9 (from eq.4b) B = 3×E - 1 = 3×2 - 1 = 6 - 1 = 5 (from eq.6a) D = 4×E + 1 = 4×2 + 1 = 8 + 1 = 9 (from eq.5a) F = 3×E = 3×2 = 6 and ABCDEF = 959926