Puzzle for October 26, 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
eq.5 may be written as: E – F = B – C – D + A which is the same as E – F = A + B – C – D In the above equation, replace A + B with C + D (from eq.2): E – F = C + D – C – D which becomes E – F = 0 Add F to both sides: E – F + F = 0 + F which makes E = F
Hint #2
Add A and F to both sides of eq.3: B – F + A + F = C – A + A + F which becomes B + A = C + F which is the same as A + B = C + F In the equation above, replace A + B with C + D (from eq.2): C + D = C + F Subtract C from both sides: C + D – C = C + F – C which means D = F
Hint #3
Subtract A from both sides of eq.2: C + D – A = A + B – A which becomes eq.2a) C + D – A = B In eq.6, substitute C + D – A for B (from eq.2a): C + D – A – A = A – C + D which becomes eq.6a) C + D – 2×A = A – C + D
Hint #4
Subtract D from both sides of eq.6a: C + D – 2×A – D = A – C + D – D which becomes C – 2×A = A – C Add C and 2×A to both sides of the above equation: C – 2×A + C + 2×A = A – C + C + 2×A which simplifies to 2×C = 3×A Divide both sides by 2: 2×C ÷ 2 = 3×A ÷ 2 which makes C = 1½×A
Hint #5
Add F to both sides of eq.3: B – F + F = C – A + F which becomes B = C – A + F Substitute E for F in the equation above: eq.3a) B = C – A + E
Hint #6
Substitute (C – A + E) for B (from eq.3a) in eq.4: A + C – (C – A + E) = E – C which is equivalent to A + C – C + A – E = E – C which becomes 2×A – E = E – C Add E and C to each side of the above equation: eq.4a) 2×A + C = 2×E
Hint #7
Substitute 1½×A for C in eq.4a: 2×A + 1½×A = 2×E which makes 3½×A = 2×E Divide both sides of the above equation by 2: 3½×A ÷ 2 = 2×E ÷ 2 which means 1¾×A = E and which also means D = F = E = 1¾×A
Hint #8
Substitute 1½×A for C, and 1¾×A for E in eq.3a: B = 1½×A – A + 1¾×A which makes B = 2¼×A
Solution
Substitute 2¼×A for B, 1½×A for C, and 1¾×A for D and E and F into eq.1: A + 2¼×A + 1½×A + 1¾×A + 1¾×A + 1¾×A = 40 which simplifies to 10×A = 40 Divide both sides of the equation above by 10: 10×A ÷ 10 = 40 ÷ 10 which means A = 4 making B = 2¼×A = 2¼ × 4 = 9 C = 1½×A = 1½ × 4 = 6 D = E = F = 1¾×A = 1¾ × 4 = 7 and ABCDEF = 496777