This puzzle showed up in several places with different anecdotes. The puzzle first specifies a decimal number with two digits after the point, such as n = 7.35. We are supposed to discover 4 different numbers such that n is equal to the sum of and the product of these 4 numbers. This means, given n, we have 2 equations with 4 unknowns:

a + b + c + d = n
a * b * c * d = n

The original puzzle gives n as 7.11.

There is an analytical solution of this problem in the Math Forum web site, posted by an author named Dr. Rob. The solution is constructed by finding the factors of the integer 711 and goes through several arguments, and finds the soluton: a = 3.16, b = 1.50, c = 1.25, and d = 1.20. The author also talks about the “uniqueness” of the solution for n = 7.11 which turns out to be true.

However, for some n values there are more than one solution. I run a Python program, and somewhat exhaustively searched for solutions (my search was not fully exhaustive but included some smart search provisions) for a, b, c, d in the range [0.01,3.99] . I am listing all solutions below that are sorted with respect to n values.

There are multiple solutions for some n values in my list, but all solutions for that n are different; for example, 7.07 has two different solutions: [0.80, 1.75, 2.02, 2.50] and [1.25, 1.00, 2.02, 2.80]. These pairs of 4 numbers indeed sum and multiply to 7.07

I also found the same (unique) solution for 7.11: [1.20, 1.25, 1.50, 3.16]

There are 47 numbers in my list and 36 of them has one solution.
4 of them has two solutions: 6.75, 7.07, 7.92, 8.10
1 of them has three solutions: 7.56

Here is the List:

6.44 [1.25, 1.60, 1.75, 1.84]
6.51 [1.25, 1.40, 1.86, 2.00]
6.63 [1.25, 1.25, 1.92, 2.21]
6.75 [1.50, 1.00, 2.25, 2.00]
6.75 [1.20, 1.25, 1.80, 2.50]
6.78 [1.13, 1.25, 2.40, 2.00]
6.84 [1.44, 1.90, 1.00, 2.50]
6.86 [1.40, 1.96, 1.00, 2.50]
7.07 [0.80, 1.75, 2.02, 2.50]
7.07 [1.25, 1.00, 2.02, 2.80]
7.08 [1.18, 1.00, 2.40, 2.50]
7.11 [1.20, 1.25, 1.50, 3.16]
7.13 [1.15, 1.00, 2.48, 2.50]
7.14 [1.02, 1.12, 2.50, 2.50]
7.20 [1.50, 1.50, 1.00, 3.20]
7.25 [0.80, 1.45, 2.50, 2.50]
7.35 [1.05, 1.00, 2.50, 2.80]
7.52 [0.64, 1.88, 2.50, 2.50]
7.56 [0.96, 1.25, 1.75, 3.60]
7.56 [1.12, 1.25, 1.44, 3.75]
7.56 [1.25, 1.25, 1.28, 3.78]
7.62 [1.27, 1.60, 1.00, 3.75]
7.65 [1.20, 1.70, 1.00, 3.75]
7.67 [0.59, 2.08, 2.50, 2.50]
7.70 [1.25, 1.60, 1.00, 3.85]
7.74 [0.80, 1.25, 2.25, 3.44]
7.82 [0.92, 1.00, 2.50, 3.40]
7.86 [0.80, 1.31, 2.00, 3.75]
7.92 [0.75, 1.25, 2.40, 3.52]
7.92 [0.90, 1.00, 2.50, 3.52]
8.01 [0.75, 1.20, 2.50, 3.56]
8.10 [0.75, 1.20, 2.40, 3.75]
8.10 [0.50, 2.40, 2.50, 2.70]
8.16 [0.85, 1.00, 2.56, 3.75]
8.22 [0.48, 2.50, 2.50, 2.74]
8.28 [0.69, 1.25, 2.50, 3.84]
8.40 [0.50, 2.40, 2.00, 3.50]
8.64 [0.50, 1.80, 2.50, 3.84]
8.67 [0.50, 1.70, 2.72, 3.75]
8.82 [0.42, 2.40, 2.50, 3.50]
8.91 [0.45, 2.50, 2.00, 3.96]
8.96 [0.40, 2.50, 2.56, 3.50]
9.00 [0.40, 2.50, 2.50, 3.60]
9.24 [0.64, 1.00, 3.75, 3.85]
9.36 [0.52, 1.25, 3.75, 3.84]
9.48 [0.50, 1.28, 3.75, 3.95]
10.8 [0.25, 3.20, 3.60, 3.75]