Puzzle for August 2, 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.4, replace B with E + F (from eq.2): C + F = E + F + E which becomes C + F = 2×E + F Subtract F from each side of the equation above: C + F - F = 2×E + F - F which makes C = 2×E
Hint #2
In eq.6, replace B with E + F (from eq.2): E + F + D = A + E + F Subtract E and F from each side of the equation above: E + F + D - E - F = A + E + F - E - F which simplifies to D = A
Hint #3
eq.1 may be re-written as: A + D + B + E + C + F = 30 In the above equation, substitute B + E for A + D (from eq.5), and B + E for C + F (from eq.4): B + E + B + E + B + E = 30 which becomes 3×(B + E) = 30 Divide both sides by 3: 3×(B + E) ÷ 3 = 30 ÷ 3 which becomes eq.1a) B + E = 10
Hint #4
Substitute A for D, and 10 for B + E (from eq.1a) in eq.5: A + A = 10 which makes 2×A = 10 Divide both sides of the above equation by 2: 2×A ÷ 2 = 10 ÷ 2 which makes A = 5 and also makes D = A = 5
Hint #5
Substitute 5 for A, and 2×E for C in eq.3: F = 5 + 2×E + E which becomes eq.3a) F = 5 + 3×E
Hint #6
Substitute 5 + 3×E for F (from eq.3a) in eq.2: B = E + 5 + 3×E which becomes eq.2a) B = 5 + 4×E
Hint #7
Substitute 5 for A and D, and 5 + 4×E for B (from eq.2a) in eq.5: 5 + 5 = 5 + 4×E + E which becomes 10 = 5 + 5×E Subtract 5 from each side of the equation above: 10 - 5 = 5 + 5×E - 5 which makes 5 = 5×E Divide both sides by 5: 5×E ÷ 5 = 5 ÷ 5 which makes E = 1
Solution
Since E = 1, then: B = 5 + 4×E = 5 + 4×1 = 5 + 4 = 9 (from eq.2a) C = 2×E = 2×1 = 2 F = 5 + 3×E = 5 + 3×1 = 5 + 3 = 8 (from eq.3a) and ABCDEF = 592518