Puzzle for September 30, 2021 ( )
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 A and D to both sides of eq.5: B – A – D + A + D = E – F + A + D which becomes B = E – F + A + D which may be written as B = A + D + E – F In the equation above, replace D + E with A + F (from eq.4): B = A + A + F – F which makes B = 2×A
Hint #2
Subtract F and C from both sides of eq.3: D + F – F – C = C + E – F – C which becomes eq.3a) D – C = E – F Subtract F and D from both sides of eq.4: A + F – F – D = D + E – F – D which becomes eq.4a) A – D = E – F
Hint #3
In eq.4a, replace E – F with D – C (from eq.3a): A – D = D – C Add D and C to both sides of the above equation: A – D + D + C = D – C + D + C which becomes eq.4b) A + C = 2×D
Hint #4
Subtract D from each side of eq.2: C + D – D = B – D which becomes C = B – D In eq.4b, substitute B – D for C: A + B – D = 2×D Add D to both sides of the equation above: A + B – D + D = 2×D + D which becomes eq.4c) A + B = 3×D
Hint #5
Substitute 2×A for B in eq.4c: A + 2×A = 3×D which becomes 3×A = 3×D Divide both sides of the above equation by 3: 3×A ÷ 3 = 3×D ÷ 3 which makes A = D
Hint #6
Substitute A for D in eq.4b: A + C = 2×A Subtract A from each side of the equation above: A + C – A = 2×A – A which makes C = A
Hint #7
Substitute A for D and C in eq.3: A + F = A + E Subtract A from both sides of the equation above: A + F – A = A + E – A which makes F = E
Hint #8
Substitute E for F, A for C and D, and 2×A for B in eq.6: E + E – A – A = A + 2×A + A + A which becomes 2×E – 2×A = 5×A Add 2×A to both sides of the above equation: 2×E – 2×A + 2×A = 5×A + 2×A which becomes 2×E = 7×A Divide both sides by 2: 2×E ÷ 2 = 7×A ÷ 2 which makes E = 3½×A and also makes F = E = 3½×A
Solution
Substitute A for C and D, 2×A for B, and 3½×A for E and F in eq.1: A + 2×A + A + A + 3½×A + 3½×A = 24 which simplifies to 12×A = 24 Divide both sides of the above equation by 12: 12×A ÷ 12 = 24 ÷ 12 which means A = 2 making B = 2×A = 2 × 2 = 4 C = D = A = 2 E = F = 3½×A = 3½ × 2 = 7 and ABCDEF = 242277