Puzzle for December 29, 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.4, replace D with A + F (from eq.2): B + A + F + E = A + C Subtract A from each side of the above equation: B + A + F + E – A = A + C – A which becomes eq.4a) B + F + E = C
Hint #2
In eq.3, replace C with B + F + E (from eq.4a): B + F + E + F = B + E – F which becomes B + 2×F + E = B + E – F In the above equation, subtract B and E from both sides, and add F to both sides: B + 2×F + E – B – E + F = B + E – F – B – E + F which simplifies to 3×F = 0 which means F = 0
Hint #3
In eq.2, substitute 0 for F: D = A + 0 which makes D = A
Hint #4
In eq.4a, substitute 0 for F: B + 0 + E = C which becomes eq.4b) B + E = C
Hint #5
Substitute B + E for C (from eq.4b), and A for D in eq.6: B + B + E – A = A + A + E which becomes 2×B + E – A = 2×A + E In the above equation, subtract E from both sides, and add A to both sides: 2×B + E – A – E + A = 2×A + E – E + A which makes 2×B = 3×A Divide both sides by 2: 2×B ÷ 2 = 3×A ÷ 2 which makes B = 1½×A
Hint #6
Substitute 0 for F, 1½×A for B, and A for D in eq.5: C + 0 – A = A + 1½×A + A which becomes C – A = 3½×A Add A to both sides of the equation above: C – A + A = 3½×A + A which makes C = 4½×A
Hint #7
Substitute 1½×A for B, A for D, and 4½×A for C in eq.4: 1½×A + A + E = A + 4½×A which becomes 2½×A + E = 5½×A Subtract 2½×A from each side of the equation above: 2½×A + E – 2½×A = 5½×A – 2½×A which makes E = 3×A
Solution
Substitute 1½×A for B, 4½×A for C, A for D, 3×A for E, and 0 for F in eq.1: A + 1½×A + 4½×A + A + 3×A + 0 = 22 which simplifies to 11×A = 22 Divide both sides of the above equation by 11: 11×A ÷ 11 = 22 ÷ 11 which means A = 2 making B = 1½×A = 1½ × 2 = 3 C = 4½×A = 4½ × 2 = 9 D = A = 2 E = 3×A = 3 × 2 = 6 and ABCDEF = 239260