Puzzle for March 11, 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.2, replace C + D + E with A + B – C (from eq.6): A + B – C + F = A + B – F Subtract A and B from both sides of the above equation: A + B – C + F – A – B = A + B – F – A – B which becomes –C + F = –F Add C and F to both sides: –C + F + C + F = –F + C + F which makes 2×F = C
Hint #2
In eq.5, replace C with 2×F: E + F = 2×F + D – F which becomes E + F = F + D Subtract F from both sides of the above equation: E + F – F = F + D – F which makes E = D
Hint #3
Add C to both sides of eq.6: A + B – C + C = C + D + E + C which becomes A + B = 2×C + D + E In the above equation, substitute D for E: A + B = 2×C + D + D which becomes eq.6a) A + B = 2×C + 2×D
Hint #4
In eq.6a, substitute A + F for B (from eq.3), and (2×F) for C: A + A + F = 2×(2×F) + 2×D which becomes 2×A + F = 4×F + 2×D Subtract F from both sides of the above equation: 2×A + F – F = 4×F + 2×D – F which becomes eq.6b) 2×A = 3×F + 2×D
Hint #5
Add A to both sides of eq.4: D + F – A + A = A – D – E + A which becomes D + F = 2×A – D – E Substitute 3×F + 2×D for 2×A (from eq.6b), and D for E in the above equation: D + F = 3×F + 2×D – D – D which becomes D + F = 3×F Subtract F from both sides: D + F – F = 3×F – F which makes D = 2×F and also makes E = D = 2×F
Hint #6
Substitute (2×F) for D in eq.6b: 2×A = 3×F + 2×(2×F) which becomes 2×A = 3×F + 4×F which makes 2×A = 7×F Divide both sides of the equation above by 2: 2×A ÷ 2 = 7×F ÷ 2 which makes A = 3½×F
Hint #7
Substitute 3½×F for A in eq.3: B = 3½×F + F which makes B = 4½×F
Solution
Substitute 3½×F for A, 4½×F for B, and 2×F for C and D and E in eq.1: 3½×F + 4½×F + 2×F + 2×F + 2×F + F = 30 which simplifies to 15×F = 30 Divide both sides of the above equation by 15: 15×F ÷ 15 = 30 ÷ 15 which means F = 2 making A = 3½×F = 3½ × 2 = 7 B = 4½×F = 4½ × 2 = 9 C = D = E = 2×F = 2 × 2 = 4 and ABCDEF = 794442