Puzzle for August 6, 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
Add C and E to both sides of eq.3: F – E + C + E = B – C + C + E which becomes F + C = B + E In eq.4, replace B + E with F + C: F + C = D + F Subtract F from both sides of the above equation: F + C – F = D + F – F which makes C = D
Hint #2
In eq.2, replace D with C: C + E = A + C Subtract C from both sides of the above equation: C + E – C = A + C – C which makes E = A
Hint #3
In eq.5, substitute C for D: C – C = A – B which becomes 0 = A – B Add B to each side of the equation above: 0 + B = A – B + B which makes B = A
Hint #4
Substitute A for E and B in eq.6: A – D = A ÷ A which becomes A – D = 1 Add D to both sides of the above equation: A – D + D = 1 + D which makes A = 1 + D and which also makes eq.6a) E = B = A = 1 + D
Hint #5
Substitute 1 + D for B and E in eq.4: 1 + D + 1 + D = D + F which becomes 2×D + 2 = D + F Subtract D from each side: 2×D + 2 – D = D + F – D which becomes eq.4a) D + 2 = F
Solution
Substitute 1 + D for A and B and E (from eq.6a), D for C, and D + 2 for F (from eq.4a) in eq.1: 1 + D + 1 + D + D + D + 1 + D + D + 2 = 47 which simplifies to 6×D + 5 = 47 Subtract 5 from both sides of the equation above: 6×D + 5 – 5 = 47 – 5 which makes 6×D = 42 Divide both sides by 6: 6×D ÷ 6 = 42 ÷ 6 which means D = 7 making A = B = E = 1 + D = 1 + 7 = 8 (from eq.6a) C = D = 7 F = 2 + D = 2 + 7 = 9 (from eq.4a) and ABCDEF = 887789