Puzzle for July 21, 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 D to both sides of eq.4: D + E + D = C – D + D which becomes eq.4a) 2×D + E = C In eq.3, replace C with 2×D + E (from eq.4a): A + D + E = 2×D + E Subtract D and E from both sides of the equation above: A + D + E – D – E = 2×D + E – D – E which makes A = D
Hint #2
In eq.5, replace C with A + D + E (from eq.3): A + D + E – A = A + B + D – F which becomes D + E = A + B + D – F In the equation above, subtract D from both sides, and add F to each side: D + E – D + F = A + B + D – F – D + F which becomes eq.5a) E + F = A + B
Hint #3
In eq.5a, replace A + B with E – F (from eq.2): E + F = E – F In the equation above, subtract E from each side, and add F to both sides: E + F – E + F = E – F – E + F which makes 2×F = 0 which means F = 0
Hint #4
Substitute D for A, and 2×D + E for C (from eq.4a) in eq.6: B – D – D + E = D + 2×D + E + D which becomes B – 2×D + E = 4×D + E In the equation above, add 2×D to both sides, and subtract E from both sides: B – 2×D + E + 2×D – E = 4×D + E + 2×D – E which makes B = 6×D
Hint #5
Substitute 0 for F, D for A, and 6×D for B in eq.5a: E + 0 = D + 6×D which makes E = 7×D
Hint #6
Substitute 7×D for E in eq.4a: 2×D + 7×D = C which makes 9×D = C
Solution
Substitute D for A, 6×D for B, 9×D for C, 7×D for E, and 0 for F in eq.1: D + 6×D + 9×D + D + 7×D + 0 = 24 which simplifies to 24×D = 24 Divide both sides of the equation above by 24: 24×D ÷ 24 = 24 ÷ 24 which means D = 1 making A = D = 1 B = 6×D = 6 × 1 = 6 C = 9×D = 9 × 1 = 9 E = 7×D = 7 × 1 = 7 and ABCDEF = 169170