Puzzle for February 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.
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Hint #1
eq.5 may be written as: eq.5a) A + D - E = B + F - A Add the left and right sides of eq.5a to the left and right sides of eq.3, respectively: E - A + (A + D - E) = A - B + F + (B + F - A) which simplifies to eq.3a) D = 2×F
Hint #2
Add E to both sides of eq.4: E + E = A + B + C - E + E which becomes eq.4a) 2×E = A + B + C
Hint #3
In eq.1, replace A + B + C with 2×E (from eq.4a), and replace D with 2×F: 2×E + 2×F + E + F = 24 which becomes 3×E + 3×F = 24 which is equivalent to 3×(E + F) = 24 Divide both sides of the above equation by 3: 3×(E + F) ÷ 3 = 24 ÷ 3 which becomes eq.1a) E + F = 8
Hint #4
eq.6 may be written as: A - B = (C + D + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × (A - B) = 3 × (C + D + F) ÷ 3 which becomes eq.6a) 3×A - 3×B = C + D + F Add B to both sides of eq.2: C - B + B = A + B + B which becomes eq.2a) C = A + 2×B
Hint #5
In eq.6a, substitute A + 2×B for C (from eq.2a), and 2×F for D: 3×A - 3×B = A + 2×B + 2×F + F which becomes 3×A - 3×B = A + 2×B + 3×F In the above equation, add 3×B to both sides, and subtract A from both sides: 3×A - 3×B + 3×B - A = A + 2×B + 3×F + 3×B - A which becomes eq.6b) 2×A = 5×B + 3×F
Hint #6
Subtract E from both sides of eq.1a: E + F - E = 8 - E which makes eq.1b) F = 8 - E Substitute (8 - E) for F (from eq.1b) in eq.6b: 2×A = 5×B + 3×(8 - E) which becomes eq.6c) 2×A = 5×B + 24 - 3×E
Hint #7
Substitute A + 2×B for C (from eq.2a) in eq.4a: 2×E = A + B + A + 2×B which becomes 2×E = 2×A + 3×B Subtract 3×B from each side of the equation above: 2×E - 3×B = 2×A + 3×B - 3×B which becomes eq.4b) 2×E - 3×B = 2×A
Hint #8
Substitute 2×E - 3×B for 2×A (from eq.4b) in eq.6c: 2×E - 3×B = 5×B + 24 - 3×E Add 3×B and 3×E to both sides of the above equation: 2×E - 3×B + 3×B + 3×E = 5×B + 24 - 3×E + 3×B + 3×E which becomes 5×E = 8×B + 24 Divide both sides by 5: 5×E ÷ 5 = (8×B + 24) ÷ 5 which becomes eq.6d) E = 1.6×B + 4.8
Hint #9
Substitute (1.6×B + 4.8) for E (from eq.6d) in eq.1b: F = 8 - (1.6×B + 4.8) which becomes F = 8 - 1.6×B - 4.8 which becomes eq.1c) F = 3.2 - 1.6×B
Hint #10
Substitute (3.2 - 1.6×B) for F (from eq.1c) in eq.3a: D = 2×(3.2 - 1.6×B) which becomes eq.5c) D = 6.4 - 3.2×B
Hint #11
Substitute (1.6×B + 4.8) for E (from eq.6d) in eq.4b: 2×(1.6×B + 4.8) - 3×B = 2×A which becomes 3.2×B + 9.6 - 3×B = 2×A which becomes 0.2×B + 9.6 = 2×A Divide both sides of the above equation by 2: (0.2×B + 9.6) ÷ 2 = 2×A ÷ 2 which becomes eq.4c) 0.1×B + 4.8 = A
Hint #12
Substitute 0.1×B + 4.8 for A (from eq.4c) in eq.2a: C = 0.1×B + 4.8 + 2×B which becomes eq.2b) C = 2.1×B + 4.8
Hint #13
Substitute 1.6×B + 4.8 for E (from eq.6d), (0.1×B + 4.8) for A (from eq.4c), and 3.2 - 1.6×B for F (from eq.1c) in eq.3: 1.6×B + 4.8 - (0.1×B + 4.8) = (0.1×B + 4.8) - B + 3.2 - 1.6×B which becomes 1.6×B + 4.8 - 0.1×B - 4.8 = -2.5×B + 8 which becomes 1.5×B = -2.5×B + 8 Add 2.5×B to both sides of the equation above: 1.5×B + 2.5×B = -2.5×B + 8 + 2.5×B which makes 4×B = 8 Divide both sides by 4: 4×B ÷ 4 = 8 ÷ 4 which makes B = 2
Solution
Since B = 2, then: A = 0.1×B + 4.8 = 0.1×2 + 4.8 = 0.2 + 4.8 = 5 (from eq.4c) C = 2.1×B + 4.8 = 2.1×2 + 4.8 = 4.2 + 4.8 = 9 (from eq.2b) D = 6.4 - 3.2×B = 6.4 - 3.2×2 = 6.4 - 6.4 = 0 (from eq.5c) E = 1.6×B + 4.8 = 1.6×2 + 4.8 = 3.2 + 4.8 = 8 (from eq.6d) F = 3.2 - 1.6×B = 3.2 - 1.6×2 = 3.2 - 3.2 = 0 (from eq.1c) and ABCDEF = 529080