Puzzle for August 22, 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.6, replace E + F with B + C (from eq.2): D + B + C = A + B + C Subtract B and C from each side of the equation above: D + B + C - B - C = A + B + C - B - C which simplifies to D = A
Hint #2
Add E and B to both sides of eq.4: C - E + E + B = E - B + E + B which becomes C + B = 2×E which may be written as eq.4a) B + C = 2×E
Hint #3
In eq.2, replace B + C with 2×E (from eq.4a): E + F = 2×E Subtract E from each side of the above equation: E + F - E = 2×E - E which makes F = E
Hint #4
In eq.5, substitute E for F: A - E = E + E which becomes A - E = 2×E Add E to both sides of the above equation: A - E + E = 2×E + E which makes A = 3×E and also makes eq.5a) A = D = 3×E
Hint #5
Subtract B from each side of eq.4a: B + C - B = 2×E - B which becomes eq.4b) C = 2×E - B
Hint #6
Substitute (2×E - B) for C (from eq.4b), and 3×E for D into eq.3: B - (2×E - B) = (2×E - B) - 3×E which becomes B - 2×E + B = -E - B which becomes 2×B - 2×E = -E - B Add 2×E and B to both sides of the above equation: 2×B - 2×E + 2×E + B = -E - B + 2×E + B which makes 3×B = E and also makes 3×B = E = F
Hint #7
Substitute (3×B) for E in eq.4b: C = 2×(3×B) - B which becomes C = 6×B - B which makes C = 5×B
Hint #8
Substitute (3×B) for E in eq.5a: A = D = 3×(3×B) which makes A = D = 9×B
Solution
Substitute 9×B for A and D, 5×B for C, and 3×B for E and F in eq.1: 9×B + B + 5×B + 9×B + 3×B + 3×B = 30 which simplifies to 30×B = 30 Divide both sides of the above equation by 30: 30×B ÷ 30 = 30 ÷ 30 which means B = 1 making A = D = 9×B = 9 × 1 = 9 C = 5×B = 5 × 1 = 5 E = F = 3×B = 3 × 1 = 3 and ABCDEF = 915933