Puzzle for January 1, 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
Add E and F to both sides of eq.5: D – F + E + F = A – E + E + F which becomes D + E = A + F In eq.3, replace A + F with D + E: eq.3a) B = D + E
Hint #2
In eq.2, replace D + E with B (from eq.3a): C = B
Hint #3
In eq.4, substitute B for C: B + D – E = A + B Subtract B from both sides of the above equation: B + D – E – B = A + B – B which becomes eq.4a) D – E = A
Hint #4
Substitute D – E for A (from eq.4a) in eq.5: D – F = D – E – E which becomes D – F = D – 2×E Subtract D from both sides of the above equation: D – F – D = D – 2×E – D which makes –F = –2×E Multiply both sides by (–1): (–1) × (–F) = (–1) × (–2×E) which makes F = 2×E
Hint #5
eq.6 may be written as: A = (C + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × A = 3 × (C + E + F) ÷ 3 which becomes eq.6a) 3×A = C + E + F
Hint #6
Substitute 2×E for F in eq.6a: 3×A = C + E + 2×E which becomes 3×A = C + 3×E Subtract 3×E from both sides of the above equation: 3×A – 3×E = C + 3×E – 3×E which makes 3×A – 3×E = C and also makes eq.6b) B = C = 3×A – 3×E
Hint #7
Substitute 3×A – 3×E for B (from eq.6b), and 2×E for F in eq.3: 3×A – 3×E = A + 2×E In the above equation, add 3×E to both sides, and subtract A from both sides: 3×A – 3×E + 3×E – A = A + 2×E + 3×E – A which becomes 2×A = 5×E Divide both sides by 2: 2×A ÷ 2 = 5×E ÷ 2 which makes A = 2½×E
Hint #8
Substitute 2½×E for A in eq.6b: B = C = 3×(2½×E) – 3×E which becomes B = C = 7½×E – 3×E which makes B = C = 4½×E
Hint #9
Substitute 2½×E for A in eq.4a: D – E = 2½×E Add E to both sides of the equation above: D – E + E = 2½×E + E which makes D = 3½×E
Solution
Substitute 2½×E for A, 4½×E for B and C, 3½×E for D, and 2×E for F in eq.1: 2½×E + 4½×E + 4½×E + 3½×E + E + 2×E = 36 which simplifies to 18×E = 36 Divide both sides of the above equation by 18: 18×E ÷ 18 = 36 ÷ 18 which means E = 2 making A = 2½×E = 2½ × 2 = 5 B = C = 4½×E = 4½ × 2 = 9 D = 3½×E = 3½ × 2 = 7 F = 2×E = 2 × 2 = 4 and ABCDEF = 599724