Puzzle for August 31, 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
Add D to both sides of eq.6: D + F + D = A – D + D which becomes eq.6a) 2×D + F = A
Hint #2
In eq.2, replace A with 2×D + F (from eq.6a): B = 2×D + F + D which becomes eq.2a) B = 3×D + F
Hint #3
In eq.5, replace B with 3×D + F (from eq.2a): 3×D + F + D = C – D which becomes 4×D + F = C – D Add D to both sides of the equation above: 4×D + F + D = C – D + D which becomes eq.5a) 5×D + F = C
Hint #4
Substitute 2×D + F for A (from eq.6a), 3×D + F for B (from eq.2a), and 5×D + F for C (from eq.5a) in eq.4: 2×D + F + 3×D + F = 5×D + F + E which becomes 5×D + 2×F = 5×D + F + E Subtract 5×D and F from both sides of the equation above: 5×D + 2×F – 5×D – F = 5×D + F + E – 5×D – F which makes F = E
Hint #5
Substitute F for E in eq.3: C = D + F + F which becomes eq.3a) C = D + 2×F
Hint #6
Substitute D + 2×F for C (from eq.3a) in eq.5a: 5×D + F = D + 2×F Subtract D and F from both sides of the above equation: 5×D + F – D – F = D + 2×F – D – F which makes 4×D = F and also makes E = F = 4×D
Hint #7
Substitute 4×D for F in eq.6a: 2×D + 4×D = A which makes 6×D = A
Hint #8
Substitute 4×D for F in eq.2a: B = 3×D + 4×D which means B = 7×D
Hint #9
Substitute 4×D for F in eq.5a: 5×D + 4×D = C which makes 9×D = C
Solution
Substitute 6×D for A, 7×D for B, 9×D for C, and 4×D for E and F in eq.1: 6×D + 7×D + 9×D + D + 4×D + 4×D = 31 which simplifies to 31×D = 31 Divide both sides of the equation above by 31: 31×D ÷ 31 = 31 ÷ 31 which means D = 1 making A = 6×D = 6×1 = 6 B = 7×D = 7×1 = 7 C = 9×D = 9×1 = 9 E = F = 4×D = 4×1 = 4 and ABCDEF = 679144