Puzzle for March 12, 2020 ( )
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.5, replace C + D + E with A + B (from eq.2): A + B – F = A – B + F Subtract A from both sides of the above equation: A + B – F – A = A – B + F – A which becomes B – F = –B + F Add F and B to each side: B – F + F + B = –B + F + F + B which becomes 2×B = 2×F Divide both sides by 2: 2×B ÷ 2 = 2×F ÷ 2 which makes B = F
Hint #2
Multiply both sides of eq.4 by (–1): (C – D) × (–1) = (D – E) × (–1) which becomes –C + D = –D + E which may be written as eq.4a) D – C = E – D
Hint #3
In eq.3, replace E – D with D – C (from eq.4a): F – C – (A – D) = D – C which is equivalent to F – C – A + D = D – C Add C and A to both sides of the equation above: F – C – A + D + C + A = D – C + C + A which becomes F + D = D + A Subtract D from both sides: F + D – D = D + A – D which makes F = A and also makes B = F = A
Hint #4
In eq.2, substitute A for B: A + A = C + D + E which becomes eq.2a) 2×A = C + D + E
Hint #5
Add C to both sides of eq.6: D + E + C = A + B + C – E + C which is the same as C + D + E = A + B + 2×C – E Substitute 2×A for C + D + E (from eq.2a), and A for B in the above equation: 2×A = A + A + 2×C – E which becomes 2×A = 2×A + 2×C – E Subtract 2×A from both sides, and add E to both sides: 2×A – 2×A + E = 2×A + 2×C – E – 2×A + E which simplifies to E = 2×C
Hint #6
Substitute 2×C for E in eq.4a: D – C = 2×C – D Add C and D to both sides of the above equation: D – C + C + D = 2×C – D + C + D which makes 2×D = 3×C Divide both sides by 2: 2×D ÷ 2 = 3×C ÷ 2 which makes D = 1½×C
Hint #7
Substitute 1½×C for D, and 2×C for E in eq.2a: 2×A = C + 1½×C + 2×C which makes 2×A = 4½×C Divide both sides of the equation above by 2: 2×A ÷ 2 = 4½×C ÷ 2 which makes A = 2¼×C and also makes B = F = A = 2¼×C
Solution
Substitute 2¼×C for A and B and F, 1½×C for D, and 2×C for E in eq.1: 2¼×C + 2¼×C + C + 1½×C + 2×C + 2¼×C = 45 which simplifies to 11¼×C = 45 Divide both sides of the equation above by 11¼: 11¼×C ÷ 11¼ = 45 ÷ 11¼ which means C = 4 making A = B = F = 2¼×C = 2¼ × 4 = 9 D = 1½×C = 1½ × 4 = 6 E = 2×C = 2 × 4 = 8 and ABCDEF = 994689