Puzzle for December 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
In eq.2, substitute (B – C + E) for F (from eq.5): B – (B – C + E) = D – E which is the same as B – B + C – E = D – E which becomes C – E = D – E Add E to both sides of the equation above: C – E + E = D – E + E which makes C = D
Hint #2
Add E and F to both sides of eq.2: B – F + E + F = D – E + E + F which becomes B + E = D + F In the above equation, replace B + E with A (from eq.3): eq.2a) A = D + F
Hint #3
eq.4 may be re-written as: D + F + E = A + B – C In the equation above, substitute A for D + F (from eq.2a): A + E = A + B – C Subtract A from each side of the equation above: A + E – A = A + B – C – A which becomes eq.4a) E = B – C
Hint #4
Substitute E for B – C (from eq.4a) in eq.5: F = E + E which becomes F = 2×E
Hint #5
Substitute D + F for A (from eq.2a) in eq.6: B + C + D – F = D + F + F which becomes B + C + D – F = D + 2×F In the equation above, add F to both sides, and subtract D from each side: B + C + D – F + F – D = D + 2×F + F – D which becomes eq.6a) B + C = 3×F
Hint #6
Add C to both sides of eq.4a: E + C = B – C + C which becomes E + C = B Substitute E + C for B, and (2×E) for F in eq.6a: E + C + C = 3×(2×E) which becomes E + 2×C = 6×E Subtract E from both sides of the above equation: E + 2×C – E = 6×E – E which makes 2×C = 5×E Divide both sides by 2: 2×C ÷ 2 = 5×E ÷ 2 which makes C = 2½×E and also makes D = C = 2½×E
Hint #7
Substitute 2½×E for C, and (2×E) for F in eq.6a: B + 2½×E = 3×(2×E) which becomes B + 2½×E = 6×E Subtract 2½×E from both sides of the above equations: B + 2½×E – 2½×E = 6×E – 2½×E which makes B = 3½×E
Hint #8
Substitute 3½×E for B in eq.3: A = 3½×E + E which makes A = 4½×E
Solution
Substitute 4½×E for A, 3½×E for B, 2½×E for C and D, and 2×E for F in eq.1: 4½×E + 3½×E + 2½×E + 2½×E + E + 2×E = 32 which simplifies to 16×E = 32 Divide both sides of the equation above by 16: 16×E ÷ 16 = 32 ÷ 16 which means E = 2 making A = 4½×E = 4½ × 2 = 9 B = 3½×E = 3½ × 2 = 7 C = D = 2½×E = 2½ × 2 = 5 F = 2×E = 2 × 2 = 4 and ABCDEF = 975524