Puzzle for June 13, 2019 ( )
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 both C and D to each side of eq.5: A - D + E + C + D = F - C + B + C + D which becomes A + E + C = F + B + D Replace B + D with E (from eq.4): A + E + C = F + E Subtract E from both sides of the above equation: A + E + C - E = F + E - E which becomes eq.5a) A + C = F
Hint #2
In eq.5a, replace F with A + B (from eq.6): A + C = A + B Subtract A from both sides of the above equation: A + C - A = A + B - A which makes C = B
Hint #3
In eq.6, substitute D + E for A (from eq.2), and C for B: F = D + E + C which may be written as F = C + D + E In eq.1, replace C + D + E with F, and A + B with F (from eq.6): F + F + F = 24 which makes 3×F = 24 Divide both sides of the equation above by 3: 3×F ÷ 3 = 24 ÷ 3 which means F = 8
Hint #4
Substitute D + E for A (from eq.2), B for C, and 8 for F in eq.5: D + E - D + E = 8 - B + B which simplifies to 2×E = 8 Divide both sides of the above equation by 2: 2×E ÷ 2 = 8 ÷ 2 which makes E = 4
Hint #5
Substitute 4 for E in eq.2: eq.2a) D + 4 = A Substitute 4 for E in eq.4: B + D = 4 Subtract D from both sides: B + D - D = 4 - D which makes B = 4 - D which also makes eq.4a) C = B = 4 - D
Solution
Substitute 8 for F, D + 4 for A (from eq.2a), and (4 - D) for B and C (from eq.4a) in eq.3: 8 - D = D + 4 - (4 - D) - (4 - D) which is the same as 8 - D = D + 4 - 4 + D - 4 + D which becomes 8 - D = 3×D - 4 Add D + 4 to both sides of the equation above: 8 - D + D + 4 = 3×D - 4 + D + 4 which simplifies to 12 = 4×D Divide both sides by 4: 12 ÷ 4 = 4×D ÷ 4 which means 3 = D making A = D + 4 = 3 + 4 = 7 (from eq.2a) C = B = 4 - D = 4 - 3 = 1 (from eq.4a) and ABCDEF = 711348