Puzzle for February 11, 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
In eq.3, replace D + E with A + B + C (from eq.5): B + C = A + B + C – F In the above equation, subtract B and C from each side, and add F to each side: B + C – B – C + F = A + B + C – F – B – C + F which simplifies to F = A
Hint #2
In eq.2, replace F with A: A + C = B + A Subtract A from each side of the equation above: A + C – A = B + A – A which becomes C = B
Hint #3
In eq.5, substitute C + D for A (from eq.4), and C for B: D + E = C + D + C + C which becomes D + E = 3×C + D Subtract D from both sides of the above equation: D + E – D = 3×C + D – D which makes E = 3×C
Hint #4
Substitute C for B, and 3×C for E in eq.6: C + C + 3×C = D – 3×C which becomes 5×C = D – 3×C Add 3×C to each side of the above equation: 5×C + 3×C = D – 3×C + 3×C which makes 8×C = D
Hint #5
Substitute 8×C for D in eq.4: C + 8×C = A which makes 9×C = A and also makes F = A = 9×C
Solution
Substitute 9×C for A and F, C for B, 8×C for D, and 3×C for E in eq.1: 9×C + C + C + 8×C + 3×C + 9×C = 31 which simplifies to 31×C = 31 Divide both sides of the above equation by 31: 31×C ÷ 31 = 31 ÷ 31 which means C = 1 making A = F = 9×C = 9 × 1 = 9 B = C = 1 D = 8×C = 8 × 1 = 8 E = 3×C = 3 × 1 = 3 and ABCDEF = 911839