Puzzle for August 29, 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
Help Area
Hint #1
Add D to both sides of eq.6: B – C + D = E – D – F + D which becomes B – C + D = E – F In eq.3, replace B – C + D with E – F: E – F = A + F Add F to each side of the above equation: E – F + F = A + F + F which becomes eq.3a) E = A + 2×F
Hint #2
Add D and A to each side of eq.5: A + B – D + D + A = D + E – A + D + A which becomes eq.5a) 2×A + B = 2×D + E Add D to both sides of eq.4: D + F + D = B – D + D which becomes eq.4a) 2×D + F = B
Hint #3
In eq.5a, replace B with 2×D + F (from eq.4a): 2×A + 2×D + F = 2×D + E Subtract 2×D from each side of the equation above: 2×A + 2×D + F – 2×D = 2×D + E – 2×D which becomes eq.5b) 2×A + F = E
Hint #4
In eq.3a, replace E with 2×A + F (from eq.5b): 2×A + F = A + 2×F Subtract A and F from both sides of the equation above: 2×A + F – A – F = A + 2×F – A – F which makes A = F
Hint #5
In eq.3a, substitute A for F: E = A + 2×A which makes eq.3b) E = 3×A
Hint #6
Substitute 3×A for E, and A for F in eq.2: 3×A – A = A + C Subtract A from both sides of the above equation: 3×A – A – A = A + C – A which makes A = C
Hint #7
Substitute A for F in eq.4a: eq.4b) 2×D + A = B
Hint #8
Substitute A for C and F in eq.3: B – A + D = A + A Add A to each side of the equation above: B – A + D + A = A + A + A which becomes eq.3c) B + D = 3×A
Hint #9
Substitute 2×D + A for B (from eq.4b) in eq.3c: 2×D + A + D = 3×A which becomes 3×D = 2×A Divide both sides of the above equation by 2: 3×D ÷ 2 = 2×A ÷ 2 which makes 1½D = A and which also makes C = F = A = 1½×D
Hint #10
Substitute (1½×D) for A in eq.3b: E = 3×(1½D) which makes E = 4½×D
Hint #11
Substitute 1½×D for F in eq.4a: 2×D + 1½×D = B which makes 3½×D = B
Solution
Substitute 1½×D for A and C and F, 3½×D for B, and 4½×D for E in eq.1: 1½×D + 3½×D + 1½×D + D + 4½×D + 1½×D = 27 which simplifies to 13½×D = 27 Divide both sides of the equation above by 13½: 13½×D ÷ 13½ = 27 ÷ 13½ which means D = 2 making A = C = F = 1½×D = 1½ × 2 = 3 B = 3½×D = 3½ × 2 = 7 E = 4½×D = 4½ × 2 = 9 and ABCDEF = 373293