Puzzle for January 28, 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 D and A to both sides of eq.3: C – D + F + D + A = E – A + D + A which becomes C + F + A = E + D which may be written as A + C + F = D + E In eq.4, replace D + E with A + C + F: A + C + F = A + F Subtract A and F from each side of the above equation: A + C + F – A – F = A + F – A – F which means C = 0
Hint #2
Add E to both sides of eq.5: E – D + E = A – E + E which becomes eq.5a) 2×E – D = A In eq.4, replace A with 2×E – D (from eq.5a): D + E = 2×E – D + F In the equation above, subtract E from each side, and add D to each side: D + E – E + D = 2×E – D + F – E + D which becomes eq.4a) 2×D = E + F
Hint #3
Add D and F to both sides of eq.2: B – D + E + D + F = D – F + D + F which becomes B + E + F = 2×D In the above equation, substitute 2×D for E + F (from eq.4a): B + 2×D = 2×D Subtract 2×D from both sides: B + 2×D – 2×D = 2×D – 2×D which makes B = 0
Hint #4
eq.6 may be written as: F = (A + B + D + E) ÷ 4 Multiply both sides of the above equation by 4: 4 × F = 4 × (A + B + D + E) ÷ 4 which becomes eq.6a) 4×F = A + B + D + E
Hint #5
eq.1 may be written as: A + B + D + E + C + F = 30 Substitute 4×F for A + B + D + E (from eq.6a), and 0 for C in the above equation: 4×F + 0 + F = 30 which becomes 5×F = 30 Divide both sides by 5: 5×F ÷ 5 = 30 ÷ 5 which makes F = 6
Hint #6
Substitute 6 for F, 2×E – D for A (from eq.5a), and 0 for B in eq.6a: 4×6 = 2×E – D + 0 + D + E which becomes 24 = 3×E Divide both sides of the equation above by 3: 24 ÷ 3 = 3×E ÷ 3 which makes 8 = E
Hint #7
Substitute 8 for E, and 6 for F in eq.4a: 2×D = 8 + 6 which makes 2×D = 14 Divide both sides of the equation above by 2: 2×D ÷ 2 = 14 ÷ 2 which makes D = 7
Solution
Substitute 8 for E, and 7 for D in eq.5a: 2×8 – 7 = A which becomes 16 – 7 = A which makes 9 = A making ABCDEF = 900786