Puzzle for February 1, 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
In eq.5, replace A + B with E (from eq.2): C + F = E + D which is the same as eq.5a) C + F = D + E
Hint #2
In eq.5a, replace D + E with F (from eq.3): C + F = F Subtract F from each side of the equation above: C + F - F = F - F which makes C = 0
Hint #3
In eq.4, substitute 0 for C: B - E = 0 + E - F which becomes B - E = E - F Add E and F to both sides of the above equation: B - E + E + F = E - F + E + F which becomes eq.4a) B + F = 2×E
Hint #4
Substitute 0 for C, and 2×E for B + F (from eq.4a) in eq.6: A + 0 = (2×E) ÷ E which makes A = 2
Hint #5
Substitute 2 for A in eq.2: eq.2a) E = 2 + B
Hint #6
Substitute (2 + B) for E (from eq.2a) in eq.4a: B + F = 2×(2 + B) which becomes B + F = 4 + 2×B Subtract B from both sides of the equation above: B + F - B = 4 + 2×B - B which makes eq.4b) F = 4 + B
Hint #7
Substitute 4 + B for F (from eq.4b), and 2 + B for E (from eq.2a) in eq.3: 4 + B = D + 2 + B Subtract 2 and B from each side of the above equation: 4 + B - 2 - B = D + 2 + B - 2 - B which makes 2 = D
Hint #8
Substitute 2 for A and D, 0 for C, 2 + B for E (from eq.2a), and 4 + B for F (from eq.4b) in eq.1: 2 + B + 0 + 2 + 2 + B + 4 + B = 25 which simplifies to 10 + 3×B = 25 Subtract 10 from both sides of the above equation: 10 + 3×B - 10 = 25 - 10 which makes 3×B = 15 Divide both sides by 3: 3×B ÷ 3 = 15 ÷ 3 which makes B = 5
Solution
Since B = 5, then: E = 2 + B = 2 + 5 = 7 (from eq.2a) F = 4 + B = 4 + 5 = 9 (from eq.4b) and ABCDEF = 252079