Puzzle for July 19, 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
In eq.4, replace B with A - E (from eq.3): A - C = A - E - F Subtract A from both sides of the above equation: A - C - A = A - E - F - A which becomes -C = -E - F Multiply both sides of the above equation by (-1): (-1) × (-C) = (-1) × (-E - F) which becomes eq.4a) C = E + F
Hint #2
In eq.2, replace E + F with C (from eq.4a): D + C = C Subtract C from both sides: D + C - C = C - C which makes D = 0
Hint #3
Add F to both sides of eq.4: A - C + F = B - F + F which becomes A - C + F = B In eq.5, substitute 0 for D, and A - C + F for B: E - F + 0 = A - C + F - C which becomes E - F = A - 2×C + F Add F to both sides of the above equation: E - F + F = A - 2×C + F + F which becomes eq.5a) E = A - 2×C + 2×F
Hint #4
Substitute (E + F) for C (from eq.4a) in eq.5a: E = A - 2×(E + F) + 2×F which is equivalent to E = A - 2×E - 2×F + 2×F which becomes E = A - 2×E Add 2×E to both sides of the above equation: E + 2×E = A - 2×E + 2×E which makes 3×E = A
Hint #5
Substitute 3×E for A in eq.3: B = 3×E - E which makes B = 2×E
Hint #6
Substitute E + F for C (from eq.4a), 3×E for A, and 2×E for B in eq.6: E + F - (3×E - F) = 3×E + 2×E which is equivalent to E + F - 3×E + F = 5×E which becomes -2×E + 2×F = 5×E Add 2×E to both sides: -2×E + 2×F + 2×E = 5×E + 2×E which means 2×F = 7×E Divide both sides by 2: 2×F ÷ 2 = 7×E ÷ 2 which means F = 3½×E
Hint #7
Substitute 3½×E for F in eq.4a: C = E + 3½×E which makes C = 4½×E
Solution
Substitute 3×E for A, 2×E for B, 4½×E for C, 0 for D, and 3½×E for F in eq.1: 3×E + 2×E + 4½×E + 0 + E + 3½×E = 28 which simplifies to 14×E = 28 Divide both sides of the equation above by 14: 14×E ÷ 14 = 28 ÷ 14 which means E = 2 making A = 3×E = 3 × 2 = 6 B = 2×E = 2 × 2 = 4 C = 4½×E = 4½ × 2 = 9 F = 3½×E = 3½ × 2 = 7 and ABCDEF = 649027