Puzzle for January 10, 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 F and B to both sides of eq.2: A – B + F + B = E – F + F + B which becomes A + F = E + B In eq.3, replace A + F with E + B: E + B = E – D Subtract E from both sides of the above equation: E + B – E = E – D – E which makes B = –D Since B and D are non-negative integers, the above equation makes: B = 0 and D = 0
Hint #2
In eq.4, replace D with 0: C – A = A – 0 Add A to each side of the equation above: C – A + A = A – 0 + A which makes C = 2×A
Hint #3
In eq.3, replace D with 0: A + F = E – 0 which becomes eq.3a) A + F = E Subtract A from both sides of eq.3a: A + F – A = E – A which becomes eq.3b) F = E – A
Hint #4
Substitute 2×A for C, and E – A for F (from eq.3b) in eq.5: E ÷ A = 2×A – A + E – A which becomes E ÷ A = E Multiply both sides of the above equation by A: E ÷ A × A = E × A which becomes E = E × A Subtract E from each side: E – E = E × A – E which becomes 0 = E × (A – 1) To make the above equation true, then: E = 0 and / or: A = 1
Hint #5
Begin checking: E = 0 ... Substituting 0 for E in eq.3a would yield: A + F = 0 Since A and F are non-negative, the equation above would make: A = 0 and F = 0
Hint #6
Finish checking: E = 0 ... Substituting 0 for B, D, E, A, and F in eq.1 would yield: 0 + 0 + C + 0 + 0 + 0 = 18 which would make C = 18 Since C is a one-digit integer, then C ≠ 18 which means E ≠ 0 and therefore means A = 1 making C = 2×A = 2 × 1 = 2
Hint #7
Substitute 2 for C, and 1 for A in eq.6: E ÷ 2 = F – 1 – 2 which becomes E ÷ 2 = F – 3 Multiply both sides of the above equation by 2: 2 × (E ÷ 2) = 2 × (F – 3) which becomes E = 2×F – 2×3 which makes eq.6a) E = 2×F – 6
Hint #8
Substitute A + F for E (from eq.3a) in eq.6a: A + F = 2×F – 6 In the above equation, subtract F from each side, and add 6 to each side: A + F – F + 6 = 2×F – 6 – F + 6 which becomes A + 6 = F Substitute 1 for A: 1 + 6 = F which makes 7 = F
Solution
Substitute 1 for A, and 7 for F in eq.3a: 1 + 7 = E which makes 8 = E making ABCDEF = 102087