Puzzle for January 20, 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.6, replace B with E + F (from eq.3): E + F + F = A + E which becomes E + 2×F = A + E Subtract E from each side of the equation above: E + 2×F – E = A + E – E which makes eq.6a) 2×F = A
Hint #2
Add E to both sides of eq.5: A – D – F + E = B – C – E + E which becomes A – D – F + E = B – C which may be written as A + E – D – F = B – C In the above equation, replace A + E with B + F (from eq.6): B + F – D – F = B – C which becomes B – D = B – C Subtract B from each side: B – D – B = B – C – B which makes –D = –C which means D = C
Hint #3
In eq.4, substitute C for D, and 2×F for A (from eq.6a): C + C + F – 2×F = 2×F + E which becomes 2×C – F = 2×F + E Add F to both sides of the equation above: 2×C – F + F = 2×F + E + F which becomes eq.4a) 2×C = E + 3×F
Hint #4
In eq.2, substitute E + F for B (from eq.3), 2×F for A (from eq.6a), and C for D: E + F + C + E = 2×F + C which becomes F + C + 2×E = 2×F + C Subtract F and C from each side of the equation above: F + C + 2×E – F – C = 2×F + C – F – C which simplifies to 2×E = F
Hint #5
Substitute (2×E) for F in eq.6a: 2×(2×E) = A which makes 4×E = A
Hint #6
Substitute 2×E for F in eq.3: E + 2×E = B which makes 3×E = B
Hint #7
Substitute (2×E) for F in eq.4a: 2×C = E + 3×(2×E) which becomes 2×C = E + 6×E which makes 2×C = 7×E Divide both sides of the above equation by 2: 2×C ÷ 2 = 7×E ÷ 2 which makes C = 3½×E and also makes D = C = 3½×E
Solution
Substitute 4×E for A, 3×E for B, 3½×E for C and D, and 2×E for F in eq.1: 4×E + 3×E + 3½×E + 3½×E + E + 2×E = 34 which simplifies to 17×E = 34 Divide each side of the equation above by 17: 17×E ÷ 17 = 34 ÷ 17 which means E = 2 making A = 4×E = 4 × 2 = 8 B = 3×E = 3 × 2 = 6 C = D = 3½×E = 3½ × 2 = 7 F = 2×E = 2 × 2 = 4 and ABCDEF = 867724