Puzzle for April 21, 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.6, replace C + F with A – B – C (from eq.3): B + A – B – C = A – C + D + E – F which becomes A – C = A – C + D + E – F In the above equation, add C and F to both sides: A – C + C + F = A – C + D + E – F + C + F which becomes A + F = A + D + E Subtract A from both sides: A + F – A = A + D + E – A which becomes eq.6a) F = D + E
Hint #2
In eq.2, replace D + E with F (from eq.6a): F = B + F Subrtact F from both sides of the equation above: F – F = B + F – F which makes 0 = B
Hint #3
In eq.3, substitute 0 for B: C + F = A – 0 – C which becomes C + F = A – C Add C to both sides of the equation above: C + F + C = A – C + C which becomes eq.3a) 2×C + F = A
Hint #4
Substitute 0 for B, and 2×C + F for A (from eq.3a) in eq.5: D + F – 0 = 2×C + F + 0 which becomes D + F = 2×C + F Subtract F from both sides of the above equation: D + F – F = 2×C + F – F which becomes D = 2×C
Hint #5
Substitute 0 for B, and 2×C + F for A (from eq.3a), and 2×C for D in eq.4: F – 0 = 2×C + F + 2×C – F which becomes F = 4×C
Hint #6
Substitute 4×C for F in eq.3a: 2×C + 4×C = A which makes 6×C = A
Hint #7
Substitute 4×C for F, and 2×C for D in eq.6a: 4×C = 2×C + E Subtract 2×C from each side of the above equation: 4×C – 2×C = 2×C + E – 2×C which makes 2×C = E
Solution
Substitute 6×C for A, 0 for B, 2×C for D and E, and 4×C for F in eq.1: 6×C + 0 + C + 2×C + 2×C + 4×C = 15 which simplifies to 15×C = 15 Divide both sides of the equation above by 15: 15×C ÷ 15 = 15 ÷ 15 which means C = 1 making A = 6×C = 6 × 1 = 6 D = E = 2×C = 2 × 1 = 2 F = 4×C = 4 × 1 = 4 and ABCDEF = 601224