Puzzle for February 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
In eq.3, replace F with A + B (from eq.2): B + C = A + A + B which becomes B + C = 2×A + B Subtract B from both sides of the above equation: B + C – B = 2×A + B – B which makes C = 2×A
Hint #2
eq.5 may be re-written as: B + C + D = A + F + E In the above equation, replace A + F with B + C (from eq.3): B + C + D = B + C + E Subtract B and C from both sides of the equation above: B + C + D – B – C = B + C + E – B – C which makes D = E
Hint #3
In eq.6, substitute D for E, and C + D for B (from eq.4): D + D – C = C + D + C which becomes 2×D – C = 2×C + D In the above equation, add C to both sides, and subtract D from both sides: 2×D – C + C – D = 2×C + D + C – D which makes eq.6a) D = 3×C
Hint #4
Substitute 2×A for C in eq.6a: D = 3×(2×A) which makes D = 6×A and also makes E = D = 6×A
Hint #5
Substitute 2×A for C, and 6×A for D in eq.4: 2×A + 6×A = B which makes 8×A = B
Hint #6
Substitute 8×A for B in eq.2: F = A + 8×A which makes F = 9×A
Solution
Substitute 8×A for B, 2×A for C, 6×A for D and E, and 9×A for F in eq.1: A + 8×A + 2×A + 6×A + 6×A + 9×A = 32 which simplifies to 32×A = 32 Divide both sides of the equation above by 32: 32×A ÷ 32 = 32 ÷ 32 which makes A = 1 making B = 8×A = 8 × 1 = 8 C = 2×A = 2 × 1 = 2 D = E = 6×A = 6 × 1 = 6 F = 9×A = 9 × 1 = 9 and ABCDEF = 182669