Puzzle for February 28, 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.
* BC and EF are 2-digit numbers (not B×C or E×F).
Scratchpad
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Hint #1
Add both C and E to each side of eq.5: A - C + C + E = B - E + C + E which becomes A + E = B + C In eq.4, replace A + E with B + C: D = B + C
Hint #2
In eq.2, replace B + C with D: D = D - F Subtract D from each side of the above equation: D - D = D - F - D which becomes 0 = -F which means 0 = F
Hint #3
In eq.3, replace F with 0: E + 0 = A + C which becomes E = A + C In eq.5, substitute (A + C) for E: A - C = B - (A + C) which is equivalent to A - C = B - A - C Add both A and C to each side of the equation above: A - C + A + C = B - A - C + A + C which simplifies to 2×A = B
Hint #4
Substitute 2×A for B (in eq.5): A - C = 2×A - E Subtract A from both sides of the equation above: A - C - A = 2×A - E - A which becomes -C = A - E Multiply both sides by (-1): -C × (-1) = (A - E) × (-1) which becomes C = -A + E which may be written as eq.5a) C = E - A
Hint #5
eq.6 may be written as: 10×B + C + D = 10×E + F Substitute (2×A) for B, E - A for C (from eq.5a), A + E for D (from eq.4), and 0 for F in the equation above: 10×(2×A) + E - A + A + E = 10×E + 0 which becomes 20×A + 2×E = 10×E Subtract 2×E from each side: 20×A + 2×E - 2×E = 10×E - 2×E which simplifies to 20×A = 8×E Divide both sides by 8: 20×A ÷ 8 = 8×E ÷ 8 which makes 2½×A = E
Hint #6
In eq.4, substitute 2½×A for E: D = A + 2½×A which makes D = 3½×A
Hint #7
In eq.5a, substitute 2½×A for E: C = 2½×A - A which makes C = 1½×A
Solution
Substitute 2×A for B, 1½×A for C, 3½×A for D, 2½×A for E, and 0 for F in eq.1: A + 2×A + 1½×A + 3½×A + 2½×A + 0 = 21 which simplifies to 10½×A = 21 Divide both sides of the equation above by 10½: 10½×A ÷ 10½ = 21 ÷ 10½ which means A = 2 making B = 2×A = 2 × 2 = 4 C = 1½×A = 1½ × 2 = 3 D = 3½×A = 3½ × 2 = 7 E = 2½×A = 2½ × 2 = 5 and ABCDEF = 243750