Puzzle for February 14, 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 (A + D - F) to each side of eq.6: B - A + (A + D - F) = C - D + F + (A + D - F) which becomes B + D - F = C + A which may also be written as B + D - F = A + C In eq.4, replace A + C with B + D - F: B + D - F = D + F Add (F - D) to each side: B + D - F + (F - D) = D + F + (F - D) which simplifies to B = 2×F
Hint #2
In eq.3, replace B with 2×F: 2×F + C - F = A + D + F which becomes F + C = A + D + F Subtract F from both sides of the equation above: F + C - F = A + D + F - F which makes eq.3a) C = A + D
Hint #3
In eq.5, substitute (A + D) for C (from eq.3a): E - D = A - (A + D) which may be written as E - D = A - A - D which becomes E - D = -D Add D to each side of the above equation: E - D + D = -D + D which means E = 0
Hint #4
In eq.4, substitute (A + D) for C (from eq.3a): A + (A + D) = D + F which becomes 2×A + D = D + F Subtract D from each side of the above equation: 2×A + D - D = D + F - D which means 2×A = F Divide both sides by 2: 2×A ÷ 2 = F ÷ 2 which means A = ½×F
Hint #5
Substitute A + D for C (from eq.3a), 2×F for B, and 0 for E into eq.2: A + D + D = 2×F + 0 + F which becomes A + 2×D = 3×F Substitute ½×F for A in the above equation: ½×F + 2×D = 3×F Subtract ½×F from each side: ½×F + 2×D - ½×F = 3×F - ½×F which simplifies to 2×D = 2½×F Divide both sides by 2: 2×D ÷ 2 = 2½×F ÷ 2 which means D = 1¼×F
Hint #6
Substitute ½×F for A, and 1¼×F for D into eq.3a: C = ½×F + 1¼×F which makes C = 1¾×F
Solution
Substitute ½×F for A, 2×F for B, 1¾×F for C, 1¼×F for D, and 0 for E in eq.1: ½×F + 2×F + 1¾×F + 1¼×F + 0 + F = 26 which simplifies to 6½×F = 26 Divide both sides by 6½: 6½×F ÷ 6½ = 26 ÷ 6½ F = 4 making A = ½×F = ½ × 4 = 2 B = 2×F = 2 × 4 = 8 C = 1¾×F = 1¾ × 4 = 7 D = 1¼×F = 1¼ × 4 = 5 and ABCDEF = 287504