Puzzle for October 13, 2020 ( )
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
In eq.6, replace A with D + E + F (from eq.4): D + E = D + E + F – D which becomes D + E = E + F Subtract E from each side of the above equation: D + E – E = E + F – E which makes D = F
Hint #2
eq.5 may be written as: B – A = A + F + E In the above equation, replace A + F with C (from eq.2): B – A = C + E Add A to both sides: B – A + A = C + E + A which becomes B = C + E + A which is the same as eq.5a) B = A + C + E
Hint #3
Substitute A + C + D for B (from eq.3) in eq.5a: A + C + D = A + C + E Subtract A and C from each side of the equation above: A + C + D – A – C = A + C + E – A – C which simplifies to D = E
Hint #4
Substitute D for E and F in eq.4: A = D + D + D which makes A = 3×D
Hint #5
Substitute 3×D for A, and D for F in eq.2: C = 3×D + D which makes C = 4×D
Hint #6
Substitute 3×D for A, and 4×D for C in eq.3: B = 3×D + 4×D + D which makes B = 8×D
Solution
Substitute 3×D for A, 8×D for B, 4×D for C, and D for E and F in eq.1: 3×D + 8×D + 4×D + D + D + D = 18 which simplifies to 18×D = 18 Divide both sides of the equation above by 18: 18×D ÷ 18 = 18 ÷ 18 which means D = 1 making A = 3×D = 3 × 1 = 3 B = 8×D = 8 × 1 = 8 C = 4×D = 4 × 1 = 4 E = F = D = 1 and ABCDEF = 384111