Puzzle for January 12, 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
In eq.4, replace A with D + E (from eq.2): B + D = D + E + F Subtract D from both sides of the above equation: B + D – D = D + E + F – D which becomes eq.4a) B = E + F
Hint #2
In eq.3, replace E + F with B (from eq.4a): C = B
Hint #3
Add A to both sides of eq.5: A + E + A = B + C – A + A which becomes eq.5a) 2×A + E = B + C In eq.6, substitute 2×A + E for B + C (from eq.5a): 2×A + E + E = A + D + F which becomes 2×A + 2×E = A + D + F Subtract A from each side of the equation above: 2×A + 2×E – A = A + D + F – A eq.6a) A + 2×E = D + F
Hint #4
Substitute D + E for A (from eq.2) in eq.6a: D + E + 2×E = D + F which becomes D + 3×E = D + F Subtract D from each side of the above equation: D + 3×E – D = D + F – D which makes 3×E = F
Hint #5
Substitute 3×E for F in eq.4a: B = E + 3×E which becomes B = 4×E which makes C = B = 4×E
Hint #6
Substitute 4×E for B and C in eq.5a: 2×A + E = 4×E + 4×E which becomes 2×A + E = 8×E Subtract E from both sides of the above equation: 2×A + E – E = 8×E – E which makes 2×A = 7×E Divide both sides by 2: 2×A ÷ 2 = 7×E ÷ 2 which makes A = 3½×E
Hint #7
Substitute 3½×E for A in eq.2: D + E = 3½×E Subtract E from each side of the above equation: D + E – E = 3½×E – E which makes D = 2½×E
Solution
Substitute 3½×E for A, 4×E for B and C, 2½×E for D, and 3×E for F in eq.1: 3½×E + 4×E + 4×E + 2½×E + E + 3×E = 36 which simplifies to 18×E = 36 Divide both sides of the equation above by 18: 18×E ÷ 18 = 36 ÷ 18 which means E = 2 making A = 3½×E = 3½ × 2 = 7 B = C = 4×E = 4 × 2 = 8 D = 2½×E = 2½ × 2 = 5 F = 3×E = 3 × 2 = 6 and ABCDEF = 788526