Puzzle for September 2, 2022 ( )
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 the left and right sides of eq.6 to the left and right sides of eq.5, respectively: B + F – A + (A – E – F) = A – D – E + (E – A – D) which becomes B – E = –2×D Add E and 2×D to both sides of the equation above: B – E + E + 2×D = –2×D + E + 2×D which becomes eq.5a) B + 2×D = E
Hint #2
In eq.4, replace E with B + 2×D (from eq.5a): B + 2×D – D = B – C + D which becomes B + D = B – C + D Subtract B and D from each side of the equation above: B + D – B – D = B – C + D – B – D which makes 0 = –C which means 0 = C
Hint #3
In eq.2, replace C with 0: B + E = A – B + 0 which becomes B + E = A – B Add B to both sides of the equation above: B + E + B = A – B + B which becomes eq.2a) 2×B + E = A
Hint #4
In eq.3, replace C with 0: D + E = 0 – E + F which becomes D + E = –E + F Add E to both sides of the above equation: D + E + E = –E + F + E which becomes eq.3a) D + 2×E = F
Hint #5
In eq.6, substitute (2×B + E) for A (from eq.2a), and (D + 2×E) for F (from eq.3a): (2×B + E) – E – (D + 2×E) = E – (2×B + E) – D which becomes 2×B + E – E – D – 2×E = E – 2×B – E – D which becomes 2×B – D – 2×E = –2×B – D Add D, 2×E, and 2×B to both sides of the above equation: 2×B – D – 2×E + D + 2×E + 2×B = –2×B – D + D + 2×E + 2×B which simplifies to 4×B = 2×E Divide both sides by 2: 4×B ÷ 2 = 2×E ÷ 2 which makes 2×B = E
Hint #6
Substitute 2×B for E in eq.2a: 2×B + 2×B = A which makes 4×B = A
Hint #7
Substitute 2×B for E in eq.5a: B + 2×D = 2×B Subtract B from each side of the equation above: B + 2×D – B = 2×B – B which makes 2×D = B Divide both sides by 2: 2×D ÷ 2 = B ÷ 2 which makes D = ½×B
Hint #8
Substitute ½×B for D, and (2×B) for E in eq.3a: ½×B + 2×(2×B) = F which becomes ½×B + 4×B = F which makes 4½×B = F
Solution
Substitute 4×B for A, 0 for C, ½×B for D, 2×B for E, and 4½×B for F in eq.1: 4×B + B + 0 + ½×B + 2×B + 4½×B = 24 which simplifies to 12×B = 24 Divide both sides of the above equation by 12: 12×B ÷ 12 = 24 ÷ 12 which means B = 2 making A = 4×B = 4 × 2 = 8 D = ½×B = ½ × 2 = 1 E = 2×B = 2 × 2 = 4 F = 4½×B = 4½ × 2 = 9 and ABCDEF = 820149