Puzzle for October 4, 2022 ( )
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.
Our thanks go out to our frequent contributor, Judah S (age 15), for providing this puzzle. Thanks, Judah!
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Hint #1
In eq.4, replace A with D + F (from eq.2): D + F + F = D + E - F which becomes D + 2×F = D + E - F In the above equation, subtract D from both sides, and add F to both sides: D + 2×F - D + F = D + E - F - D + F which simplifies to 3×F = E
Hint #2
In eq.1, replace E with 3×F: 3×F = C - F Add F to both sides of the above equation: 3×F + F = C - F + F which makes 4×F = C
Hint #3
In eq.3, substitute 4×F for C, and D + F for A (from eq.2): 4×F = D + F + D which makes 4×F = 2×D + F Subtract F from each side of the above equation: 4×F - F = 2×D + F - F which makes 3×F = 2×D Divide both sides by 2: 3×F ÷ 2 = 2×D ÷ 2 which makes 1½×F = D
Hint #4
Substitute 1½×F for D in eq.2: A = 1½×F + F which makes A = 2½×F
Hint #5
Substitute 3×F for E, and 1½×F for D in eq.5: F = 3×F ÷ 1½×F which makes F = 2 and makes A = 2½×F = 2½ × 2 = 5 C = 4×F = 4 × 2 = 8 D = 1½×F = 1½ × 2 = 3 E = 3×F = 3 × 2 = 6
Solution
Substitute 8 for C, 2 for F, 5 for A, and 3 for D in eq.6: B = (8 × 2) - (5 × 3) which becomes B = 16 - 15 which makes B = 1 and makes ABCDEF = 518362