Puzzle for June 19, 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
Add A and F to both sides of eq.3: F – A + A + F = A + C – F + A + F which becomes eq.3a) 2×F = 2×A + C Add F to both sides of eq.6: B – C + F + F = A + C + E – F + F which becomes eq.6a) B – C + 2×F = A + C + E
Hint #2
In eq.6a, replace 2×F with 2×A + C (from eq.3a): B – C + 2×A + C = A + C + E which becomes B + 2×A = A + C + E Subtract A from both sides of the above equation: B + 2×A – A = A + C + E – A which becomes B + A = C + E which may be written as eq.6b) A + B = C + E
Hint #3
In eq.6b, replace A + B with C + D (from eq.2): C + D = C + E Subtract C from each side of the above equation: C + D – C = C + E – C which makes D = E
Hint #4
Add A and C to both sides of eq.5: D + E – A – C + A + C = C + F + A + C which becomes eq.5a) D + E = 2×C + F + A Add A and F to both sides of eq.4: B + D – A – F + A + F = A + C + A + F which becomes eq.4a) B + D = 2×A + C + F
Hint #5
Subtract the left and right sides of eq.5a from the left and right sides of eq.4a, respectively: B + D – (D + E) = 2×A + C + F – (2×C + F + A) which becomes B + D – D – E = 2×A + C + F – 2×C – F – A which becomes eq.4b) B – E = A – C
Hint #6
Subtract B and C from both sides of eq.2: C + D – B – C = A + B – B – C which becomes D – B = A – C In eq.4b, substitute D – B for A – C, and D for E: B – D = D – B Add D and B to both sides of the above equation: B – D + D + B = D – B + D + B which makes 2×B = 2×D Divide both sides by 2: 2×B ÷ 2 = 2×D ÷ 2 which makes B = D
Hint #7
Substitute B for D in eq.2: C + B = A + B Subtract B from both sides of the equation above: C + B – B = A + B – B which makes C = A
Hint #8
Substitute A for C in eq.3a: 2×F = 2×A + A which becomes 2×F = 3×A Divide both sides of the above equation by 2: 2×F ÷ 2 = 3×A ÷ 2 which makes F = 1½×A
Hint #9
Substitute D for B, A for C, and 1½×A for F in eq.4a: D + D = 2×A + A + 1½×A which becomes 2×D = 4½×A Divide both sides of the equation above by 2: 2×D ÷ 2 = 4½×A ÷ 2 which makes D = 2¼×A and also makes B = D = E = 2¼×A
Solution
Substitute 2¼×A for B and D and E, A for C, and 1½×A for F in eq.1: A + 2¼×A + A + 2¼×A + 2¼×A + 1½×A = 41 which simplifies to 10¼×A = 41 Divide both sides of the above equation by 10¼: 10¼×A ÷ 10¼ = 41 ÷ 10¼ which means A = 4 making B = D = E = 2¼×A = 2¼ × 4 = 9 C = A = 4 F = 1½×A = 1½ × 4 = 6 and ABCDEF = 494996