Puzzle for May 3, 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
Add E to both sides of eq.2: C + D + E = B – E + E which becomes eq.2a) C + D + E = B Add E to both sides of eq.4: D – C + E = A – E + E which becomes eq.4a) D – C + E = A
Hint #2
In eq.3, replace A with D – C + E (from eq.4a), and B with C + D + E (from eq.2a): D – C + E + C = C + D + E + E which becomes D + E = C + D + 2×E Subtract D and E from each side of the equation above: D + E – D – E = C + D + 2×E – D – E which simplifies to eq.3a) 0 = C + E
Hint #3
Since C and E must be non-negative, eq.3a makes: C = 0 and E = 0
Hint #4
Substitute 0 for C and E in eq.2: 0 + D = B – 0 which makes D = B
Hint #5
Substitute 0 for C and E in eq.3: A + 0 = B + 0 which makes A = B and also makes A = B = D
Hint #6
Substitute A for B and D, and 0 for C in eq.5: F = (A + 0 + A) ÷ A which becomes F = 2×A ÷ A which makes F = 2
Solution
Substitute A for B and D, 0 for C and E, and 2 for F in eq.1: A + A + 0 + A + 0 + 2 = 29 which becomes 3×A + 2 = 29 Subtract 2 from each side of the above equation: 3×A + 2 – 2 = 29 – 2 which makes 3×A = 27 Divide both sides by 3: 3×A ÷ 3 = 27 ÷ 3 which means A = 9 making B = D = A = 9 and ABCDEF = 990902