Puzzle for June 11, 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
Add C and F to both sides of eq.6: B – C + C + F = A – F + C + F which becomes B + F = A + C In eq.2, replace A + C with B + F: D + F = B + F Subtract F from each side of the equation above: D + F – F = B + F – F which makes D = B
Hint #2
In eq.3, replace B with D: A + D + D = C + E which becomes A + 2×D = C + E Subtract A and E from each side of the above equation: A + 2×D – A – E = C + E – A – E which becomes eq.3a) 2×D – E = C – A
Hint #3
In eq.5, substitute 2×D – E for C – A (from eq.3a): 2×D – E = D – E + F In the above equation, add E to both sides, and subtract D from each side: 2×D – E + E – D = D – E + F + E – D which simplifies to D = F
Hint #4
Substitute D for both B and F in eq.4: D – E = D – C Subtract D from each side of the equation above: D – E – D = D – C – D –E = –C which means E = C
Hint #5
Substitute C for E, and D for F in eq.5: C – A = D – C + D which becomes C – A = 2×D – C Add C to each side of the equation above: C – A + C = 2×D – C + C which becomes eq.5a) 2×C – A = 2×D
Hint #6
Substitute D for B and F in eq.6: D – C = A – D Add C and D to each side of the above equation: D – C + C + D = A – D + C + D which becomes eq.6a) 2×D = A + C
Hint #7
Substitute A + C for 2×D (from eq.6a) in eq.5a: 2×C – A = A + C In the equation above, add A to each side, and subtract C from both sides: 2×C – A + A – C = A + C + A – C which makes C = 2×A and also makes E = C = 2×A
Hint #8
Substitute 2×A for C in eq.6a: 2×D = A + 2×A which makes 2×D = 3×A Divide both sides of the equation above by 2: 2×D ÷ 2 = 3×A ÷ 2 which makes D = 1½×A and also makes B = F = D = 1½×A
Solution
Substitute 1½×A for B and D and F, and 2×A for C and E in eq.1: A + 1½×A + 2×A + 1½×A + 2×A + 1½×A = 38 which simplifies to 9½×A = 38 Divide both sides of the equation above by 9½: 9½×A ÷ 9½ = 38 ÷ 9½ which means A = 4 making B = D = F = 1½×A = 1½ × 4 = 6 C = E = 2×A = 2 × 4 = 8 and ABCDEF = 468686