Puzzle for October 6, 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 the left and right sides of eq.2 to the left and right sides of eq.4, respectively: E - C + C + F = A + F + B + D which becomes E + F = A + F + B + D Subtract F from both sides of the above equation: E + F - F = A + F + B + D - F which becomes eq.4a) E = A + B + D
Hint #2
In eq.5, replace E with A + B + D (from eq.4a): B + C = A + B + D - D which becomes B + C = A + B Subtract B from each side of the equation above: B + C - B = A + B - B which makes C = A
Hint #3
In eq.3, replace A with C: eq.3a) C + B = D + F
Hint #4
Subtract the left and right sides of eq.3a from the left and right sides of eq.2, respectively: C + F - (C + B) = B + D - (D + F) which becomes C + F - C - B = B + D - D - F which becomes F - B = B - F Add B and F to both sides of the above equation: F - B + B + F = B - F + B + F which becomes 2×F = 2×B Divide both sides by 2: 2×F ÷ 2 = 2×B ÷ 2 which makes F = B
Hint #5
In eq.3, substitute B for F: A + B = D + B Subtract B from each side of the above equation: A + B - B = D + B - B which makes A = D
Hint #6
Substitute B for F, and A for both C and D in eq.6: B + B = A + A + A + E which becomes 2×B = 3×A + E Subtract 3×A from both sides of the equation above: 2×B - 3×A = 3×A + E - 3×A which becomes eq.6a) 2×B - 3×A = E
Hint #7
Substitute 2×B - 3×A for E (from eq.6a), and A for D in eq.4a: 2×B - 3×A = A + B + A which makes 2×B - 3×A = 2×A + B In the above equation, add 3×A to both sides, and subtract B from both sides: 2×B - 3×A + 3×A - B = 2×A + B + 3×A - B which makes B = 5×A and also makes F = B = 5×A
Hint #8
Substitute 5×A for B in eq.6a: 2(5×A) - 3×A = E which becomes 10×A - 3×A = E which makes 7×A = E
Solution
Substitute 5×A for B and F, A for C and D, and 7×A for E in eq.1: A + 5×A + A + A + 7×A + 5×A = 20 which simplifies to 20×A = 20 Divide both sides of the above equation by 20: 20×A ÷ 20 = 20 ÷ 20 which means A = 1 making B = F = 5×A = 5 × 1 = 5 C = D = A = 1 E = 7×A and ABCDEF = 151175