Puzzle for September 14, 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 positive integer.
Scratchpad
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Hint #1
Add D to both sides of eq.5: D + E + D = B + C – D + F + D which becomes eq.5a) 2×D + E = B + C + F Add C + E to both sides of eq.2: B – E + F + C + E = C – D + C + E which becomes B + F + C = 2×C – D + E which may be written as eq.2a) B + C + F = 2×C – D + E
Hint #2
In eq.5a, replace B + C + F with 2×C – D + E (from eq.2a): 2×D + E = 2×C – D + E In the above equation, add D to both sides, and subtract E from both sides: 2×D + E + D – E = 2×C – D + E + D – E which simplifies to eq.5b) 3×D = 2×C Divide both sides by 2: 3×D ÷ 2 = 2×C ÷ 2 which means 1½×D = C
Hint #3
Add F to each side of eq.3: E – F + F = D + F + F which becomes E = D + 2×F In eq.5a, replace E with D + 2×F: 2×D + D + 2×F = B + C + F Subtract F from each side of the equation above: 2×D + D + 2×F – F = B + C + F – F which becomes eq.3a) 3×D + F = B + C
Hint #4
In eq.3a, substitute 2×C for 3×D (from eq.5b): 2×C + F = B + C Subtract C from each side of the equation above: 2×C + F – C = B + C – C which becomes C + F = B Substitute A for C + F (from eq.1): A = B
Hint #5
Substitute A for B in eq.4: A + C = A + E Subtract A from each side of the equation above: A + C – A = A + E – A which means C = E and which makes C = E = 1½×D
Hint #6
Substitute C for E in eq.6: C = C × F Divide each side of the above equation by C (implies C ≠ 0): C ÷ C = C × F ÷ C which makes 1 = F
Hint #7
Substitute 1½×D for E, and 1 for F in eq.3: 1½×D – 1 = D + 1 Add 1 to both sides, and subtract D from each side: 1½×D – 1 + 1 – D = D + 1 + 1 – D which becomes ½×D = 2 Multiply both sides by 2: ½×D × 2 = 2 × 2 which makes D = 4 making C = E = 1½×D = 1½ × 4 = 6
Solution
Substitute 6 for C, and 1 for F in eq.1: 6 + 1 = A which makes 7 = A making B = A = 7 and ABCDEF = 776461