Puzzle for August 2, 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 A and C to both sides of eq.2: E – C + A + C = B + D – A + A + C which becomes eq.2a) E + A = B + D + C
Hint #2
Add C, D, and F to both sides of eq.5: B – F + C + D + F = E – C – D + C + D + F which becomes B + C + D = E + F which may be written as B + D + C = E + F Substitute E + F for B + D + C in eq.2a: E + A = E + F Subtract E from both sides of the above equation: E + A – E = E + F – E which makes A = F
Hint #3
Add C and E to both sides of eq.4: A – E + F + C + E = E – C + C + E which becomes eq.4a) A + F + C = 2×E
Hint #4
eq.1 may be written as: A + F + C + B + D + E = 33 Substitute 2×E for A + F + C (from eq.4a), and E – B for B + D (from eq.3) in the above equation: 2×E + E – B + E = 33 which becomes 4×E – B = 33 Add B to both sides, and subtract 33 from each side: 4×E – B + B – 33 = 33 + B – 33 which becomes eq.1a) 4×E – 33 = B
Hint #5
To make eq.1a true, check several possible values for E and B: If E = 9, then B = 4×9 – 33 = 36 – 33 = 3 If E = 8, then B = 4×8 – 33 = 32 – 33 = –1 If E < 8, then B < –1 Since B must be a non-negative integer, then B = 3 which makes E = 9
Hint #6
Substitute 3 for B, and 9 for E in eq.3: 3 + D = 9 – 3 which means 3 + D = 6 Subtract 3 from both sides of the above equation: 3 + D – 3 = 6 – 3 which makes D = 3
Hint #7
Substitute 9 for E, 3 for B, and 3 for D in eq.2: 9 – C = 3 + 3 – A which becomes 9 – C = 6 – A In the above equation, add A and C to both sides, and subtract 6 from both sides: 9 – C + A + C – 6 = 6 – A + A + C – 6 which simplifies to eq.2b) 3 + A = C
Solution
Substitute 9 for E, A for F, and (3 + A) for C (from eq.2b) in eq.4: A – 9 + A = 9 – (3 + A) which becomes 2×A – 9 = 9 – 3 – A which becomes 2×A – 9 = 6 – A Add both 9 and A to each side of the above equation: 2×A – 9 + 9 + A = 6 – A + 9 + A which makes 3×A = 15 Divide both sides by 3: 3×A ÷ 3 = 15 ÷ 3 which means A = 5 making C = A + 3 = 5 + 3 = 8 (from eq.2b) F = A = 5 and ABCDEF = 538395