Puzzle for December 12, 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
Help Area
Hint #1
In eq.4, substitute (B – C + F) for D (from eq.5): F – (B – C + F) = C – A which is equivalent to F – B + C – F = C – A which becomes –B + C = C – A Subtract C from both sides of the above equation: –B + C – C = C – A – C which makes –B = –A which means B = A
Hint #2
In eq.3, replace B with A: E – D = C – A In eq.4, replace C – A with E – D: F – D = E – D Add D to both sides of the equation above: F – D + D = E – D + D which makes F = E
Hint #3
In eq.2, substitute F for E: A = D + F + F which becomes eq.2a) A = D + 2×F
Hint #4
In eq.4, substitite (D + 2×F) for A (from eq.2a): F – D = C – (D + 2×F) which is the same as F – D = C – D – 2×F Add 2×F and D to each side of the equation above: F – D + 2×F + D = C – D – 2×F + 2×F + D which simplifies to eq.4a) 3×F = C
Hint #5
In eq.6, substitute A for B, and subtract D from both sides: A + D – D = C – D – D which becomes eq.6a) A = C – 2×D
Hint #6
Substitute D + 2×F for A (from eq.2a), and 3×F for C in eq.6a: D + 2×F = 3×F – 2×D In the equation above, add 2×D to both sides, and subtract 2×F from each side: D + 2×F + 2×D – 2×F = 3×F – 2×D + 2×D – 2×F which simplifies to 3×D = F and also means E = F = 3×D
Hint #7
Substitute (3×D) for F in eq.4a: 3×(3×D) = C which makes 9×D = C
Hint #8
Substitute 9×D for C in eq.6a: A = 9×D – 2×D which makes A = 7×D and also makes B = A = 7×D
Solution
Substitute 7×D for A and B, 9×D for C, and 3×D for E and F in eq.1: 7×D + 7×D + 9×D + D + 3×D + 3×D = 30 which simplifies to 30×D = 30 Divide both sides of the equation above by 30: 30×D ÷ 30 = 30 ÷ 30 which means D = 1 making A = B = 7×D = 7 × 1 = 7 C = 9×D = 9 × 1 = 9 E = F = 3×D = 3 × 1 = 3 and ABCDEF = 779133