Puzzle for May 25, 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 B + C with E - B (from eq.4): E + F = A + E - B In the equation above, subtract E from both sides, and add B to both sides: E + F - E + B = A + E - B - E + B which becomes eq.5a) F + B = A
Hint #2
In eq.2, replace A with F + B (from eq.5a): F + B = C + D Subtract B and C from both sides of the equation above: F + B - B - C = C + D - B - C which becomes eq.2a) F - C = D - B
Hint #3
Add the left and right sides of eq.2a to the left and right sides of eq.3, respectively: C + F + F - C = B + D + D - B which makes 2×F = 2×D Divide both sides of the above equation by 2: 2×F ÷ 2 = 2×D ÷ 2 which makes F = D
Hint #4
In eq.3, substitute D for F : C + D = B + D Subtract D from each side of the equation above: C + D - D = B + D - D which makes C = B
Hint #5
Substitute B for C in eq.4: B + B = E - B which becomes 2×B = E - B Add B to both sides of the equation above: 2×B + B = E - B + B which makes 3×B = E
Hint #6
Substitute 3×B for E, C + D for A (from eq.2), and D for F in eq.6: B + C + 3×B = C + D + D + D which becomes 4×B + C = C + 3×D Subtract C from each side of the above equation: 4×B + C - C = C + 3×D - C which makes 4×B = 3×D Divide both sides by 3: 4×B ÷ 3 = 3×D ÷ 3 which makes 1⅓×B = D and also makes 1⅓×B = D = F
Hint #7
Substitute B for C, and 1⅓×B for F in eq.5a: 1⅓×B + B = A which makes 2⅓×B = A
Solution
Substitute 2⅓×B for A, B for C, 1⅓×B for D and F, and 3×B for E in eq.1: 2⅓×B + B + B + 1⅓×B + 3×B + 1⅓×B = 30 which simplifies to 10×B = 30 Divide both sides of the above equation by 10: 10×B ÷ 10 = 30 ÷ 10 which means B = 3 making A = 2⅓×B = 2⅓ × 3 = 7 C = B = 3 D = F = 1⅓×B = 1⅓ × 3 = 4 E = 3×B = 3 × 3 = 9 and ABCDEF = 733494