Puzzle for July 16, 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
Subtract E from both sides of eq.2: B + E – E = A + D – E which becomes B = A + D – E In eq.4, replace B with A + D – E: A + D – E – D + F = C + D which becomes eq.4a) A – E + F = C + D
Hint #2
In eq.3, replace C + D with A – E + F (from eq.4a): A – E + F = A + E + F Subtract A, E, and F from both sides of the above equation: A – E + F – A – E – F = A + E + F – A – E – F which simplifies to –2×E = 0 which means E = 0
Hint #3
In eq.2, substitute 0 for E: B + 0 = A + D which becomes eq.2a) B = A + D
Hint #4
In eq.5, substitute 0 for E: C + 0 – A = A – C + F which becomes C – A = A – C + F In the above equation, subtract A from each side, and add C to each side: C – A – A + C = A – C + F – A + C which becomes eq.5a) 2×C – 2×A = F
Hint #5
Substitute 0 for E, and 2×C – 2×A for F (from eq.5a) in eq.4a: A – 0 + 2×C – 2×A = C + D which becomes 2×C – A = C + D Subtract C from both sides of the equation above: 2×C – A – C = C + D – C which becomes eq.4b) C – A = D
Hint #6
Substitute C – A for D (from eq.4b) in eq.2a: B = A + C – A which makes B = C
Hint #7
Substitute B for C in eq.4: B – D + F = B + D In the above equation, subtract B from both sides, and add D to both sides: B – D + F – B + D = B + D – B + D which makes F = 2×D
Hint #8
Substitute 2×D for F, A + D for B (from eq.2a), and 0 for E in eq.6: D + 2×D = A + A + D – D + 0 – 2×D which becomes 3×D = 2×A – 2×D Add 2×D to both sides of the above equation: 3×D + 2×D = 2×A – 2×D + 2×D which becomes 5×D = 2×A Divide both sides by 2: 5×D ÷ 2 = 2×A ÷ 2 which makes 2½×D = A
Hint #9
Substitute 2½×D for A in eq.2a: B = 2½×D + D which makes B = 3½×D and also makes C = B = 3½×D
Solution
Substitute 2½×D for A, 3½×D for B and C, 0 for E, and 2×D for F in eq.1: 2½×D + 3½×D + 3½×D + D + 0 + 2×D = 25 which simplifies to 12½×D = 25 Divide both sides of the above equation by 12½: 12½×D ÷ 12½ = 25 ÷ 12½ which means D = 2 making A = 2½×D = 2½ × 2 = 5 B = C = 3½×D = 3½ × 2 = 7 F = 2×D = 2 × 2 = 4 and ABCDEF = 577204