Puzzle for June 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
Multiply both sides of eq.5 by (–1): (F – E) × (–1) = (C – D) × (–1) which becomes –F + E = –C + D which may be written as E – F = D – C In eq.4, replace E – F with D – C: D – C = A – C Add C to both sides of the above equation: D – C + C = A – C + C which makes D = A
Hint #2
In eq.3, replace D with A: B = A – A which means B = 0
Hint #3
In eq.2, substitute A for D, and 0 for B: A + C + A = 0 + E + F which becomes 2×A + C = E + F which may be written as eq.2a) E + F = 2×A + C
Hint #4
Add the left and right sides of eq.2a to the left and right sides of eq.4, respectively: E – F + E + F = A – C + 2×A + C which becomes 2×E = 3×A Divide both sides of the above equation by 2: 2×E ÷ 2 = 3×A ÷ 2 which makes E = 1½×A
Hint #5
Substitute A for D, 1½×A for E, and 0 for B in eq.6: A + 1½×A = 0 + F which makes 2½×A = F
Hint #6
Substitute 1½×A for E, and 2½×A for F in eq.2a: 1½×A + 2½×A = 2×A + C which becomes 4×A = 2×A + C Subtract 2×A from each side of the equation above: 4×A – 2×A = 2×A + C – 2×A which makes 2×A = C
Solution
Substitute 0 for B, 2×A and C, A for D, 1½×A for E, and 2½×A for F in eq.1: A + 0 + 2×A + A + 1½×A + 2½×A = 16 which simplifies to 8×A = 16 Divide both sides of the equation above by 8: 8×A ÷ 8 = 16 ÷ 8 which means A = 2 making C = 2×A = 2 × 2 = 4 D = A = 2 E = 1½×A = 1½ × 2 = 3 F = 2½×A = 2½ × 2 = 5 and ABCDEF = 204235