Puzzle for October 5, 2021 ( )
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 D with B + F (from eq.3): B + F + F = B + E which becomes B + 2×F = B + E Subtract B from each side of the above equation: B + 2×F – B = B + E – B which makes 2×F = E
Hint #2
In eq.6, replace E with 2×F, and A + B with C (from eq.2): 2×F + F = C + C which makes 3×F = 2×C Divide both sides of the above equation by 2: 3×F ÷ 2 = 2×C ÷ 2 which makes 1½×F = C
Hint #3
In eq.4, substitute B + F for D (from eq.3), 1½×F for C, and 2×F for E: B + B + F = 1½×F + 2×F which becomes 2×B + F = 3½×F Subtract F from each side of the equation above: 2×B + F – F = 3½×F – F which makes 2×B = 2½×F Divide both sides by 2: 2×B ÷ 2 = 2½×F ÷ 2 which makes B = 1¼×F
Hint #4
Substitute 1¼×F for B in eq.3: D = 1¼×F + F which makes D = 2¼×F
Hint #5
Substitute 1½×F for C, and 1¼×F for B in eq.2: 1½×F = A + 1¼×F Subtract 1¼×F from both sides of the above equation: 1½×F – 1¼×F = A + 1¼×F – 1¼×F which makes ¼×F = A
Solution
Substitute ¼×F for A, 1¼×F for B, 1½×F for C, 2¼×F for D, and 2×F for E in eq.1: ¼×F + 1¼×F + 1½×F + 2¼×F + 2×F + F = 33 which simplifies to 8¼×F = 33 Divide both sides of the above equation by 8¼: 8¼×F ÷ 8¼ = 33 ÷ 8¼ which means F = 4 making A = ¼×F = ¼ × 4 = 1 B = 1¼×F = 1¼ × 4 = 5 C = 1½×F = 1½ × 4 = 6 D = 2¼×F = 2¼ × 4 = 9 E = 2×F = 2 × 4 = 8 and ABCDEF = 156984