Puzzle for March 11, 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
eq.2 may be written as: A + C + B = D + E In the above equation, replace A + C with B + E (from eq.3): B + E + B = D + E which becomes 2×B + E = D + E Subtract E from each side of the above equation: 2×B + E - E = D + E - E which makes eq.2a) 2×B = D
Hint #2
In eq.5, replace D with 2×B: 2×B - B = B - E which becomes B = B - E Subtract B from both sides of the equation above: B - B = B - E - B which makes 0 = -E which means 0 = E
Hint #3
In eq.6, substitute A + B + C for D + E (from eq.2): A + B + F = A + B + C - F Subtract both A and B from each side of the above equation: A + B + F - A - B = A + B + C - F - A - B which simplifies to F = C - F Add F to both sides: F + F = C - F + F which makes 2×F = C
Hint #4
Substitute 2×F for C, and 0 for E in eq.4: 2×F - 0 + F = B - F which makes 3×F = B - F Add F to both sides of the above equation: 3×F + F = B - F + F which makes 4×F = B
Hint #5
Substitute (4×F) for B in eq.2a: 2×(4×F) = D which makes 8×F = D
Hint #6
Substitute 2×F for C, 4×F for B, and 0 for E in eq.3: A + 2×F = 4×F + 0 Subtract 2×F from each side: A + 2×F - 2×F = 4×F + 0 - 2×F which makes A = 2×F
Solution
Substitute 2×F for A and C, 4×F for B, 8×F for D, and 0 for E in eq.1: 2×F + 4×F + 2×F + 8×F + 0 + F = 17 which becomes 17×F = 17 Divide both sides by 17: 17×F ÷ 17 = 17 ÷ 17 which means F = 1 making A = C = 2×F = 2 × 1 = 2 B = 4×F = 4 × 1 = 4 D = 8×F = 8 × 1 = 8 and ABCDEF = 242801