Puzzle for June 30, 2022 ( )
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
eq.4 may be written as: B + F + C – A – D = A – B + D In the above equation, replace B + F with A + D (from eq.2): A + D + C – A – D = A – B + D which becomes C = A – B + D which may be written as eq.4a) C = A + D – B
Hint #2
In eq.4a, replace A + D with B + F (from eq.2): C = B + F – B which makes C = F
Hint #3
In eq.5, substitute C for F: A – D = C ÷ C which becomes A – D = 1 In the above equation, add D to both sides, and subtract 1 from both sides: A – D + D – 1 = 1 + D – 1 which becomes eq.5a) A – 1 = D
Hint #4
Substitute A – 1 for D (from eq.5a) into eq.3: (A – 1) – E = A – (A – 1) which becomes A – 1 – E = A – A + 1 which becomes A – 1 – E = 1 In the above equation, add E to both sides, and subtract 1 from both sides: A – 1 – E + E – 1 = 1 + E – 1 which becomes eq.3a) A – 2 = E
Hint #5
Substitute (A – 2) for E (from eq.3a), and (A – 1) for D (from eq.5a) in eq.6: A + B – (A – 2) = (A + (A – 2)) ÷ (A – 1) which becomes A + B – A + 2 = (2×A – 2) ÷ (A – 1) which becomes eq.6a) B + 2 = (2×A – 2) ÷ (A – 1)
Hint #6
Multiply both sides of eq.6a by (A – 1): (A – 1) × (B + 2) = (A – 1) × (2×A – 2) ÷ (A – 1) which becomes A×B + 2×A – B – 2 = 2×A – 2 In the above equation, subtract 2×A from both sides, and add B and 2 to both sides: A×B – B + 2×A – 2 – 2×A + B + 2 = 2×A – 2 – 2×A + B + 2 which simplifies to A×B = B To make the above equation true, either: A = 1 or B = 0
Hint #7
Check: A = 1 ... Substituting 1 for A in eq.5a would yield: 1 – 1 = D which would make 0 = D However, D ≠ 0 (from eq.6), therefore: A ≠ 1 making B = 0
Hint #8
Substitute 0 for B, and A – 1 for D (from eq.5a) into eq.2: 0 + F = A + A – 1 which makes F = 2×A – 1 and also makes eq.2a) C = F = 2×A – 1
Hint #9
Substitute 0 for B, 2×A – 1 for C and F (from eq.2a), A – 1 for D (from eq.5a), and A – 2 for E (from eq.3a) in eq.1: A + 0 + 2×A – 1 + A – 1 + A – 2 + 2×A – 1 = 30 which simplifies to 7×A – 5 = 30 Add 5 to both sides of the above equation: 7×A – 5 + 5 = 30 + 5 which makes 7×A = 35 Divide both sides by 7: 7×A ÷ 7 = 35 ÷ 7 which makes A = 5
Solution
Since A = 5, then C = F = 2×A – 1 = 2×5 – 1 = 10 – 1 = 9 (from eq.2a) D = A – 1 = 5 – 1 = 4 (from eq.5a) E = A – 2 = 5 – 2 = 3 (from eq.3a) and ABCDEF = 509439