Dartboard Totals

I've only made one post about darts and dartboards and that was Measuring Dartsmanship on Wednesday 24th of January 2024. In this post, I want to quantify the number the of ways in which a certain total can be achieved using one, two or three darts. To this end, I developed a program in SageMath to do the job and at first glance it worked fine. Figure 1 shows the code using 75 as sample input (permalink).

Figure 1

The output is shown below (there are 194 possible ways of achieving a total of 75):

[[1, 14, 60], [1, 17, 57], [1, 20, 54], [1, 26, 48], [1, 34, 40], [1, 36, 38], [1, 42, 32], [2, 13, 60], [2, 16, 57], [2, 22, 51], [2, 28, 45], [2, 33, 40], [2, 39, 34], [2, 54, 19], [3, 12, 60], [3, 15, 57], [3, 18, 54], [3, 21, 51], [3, 24, 48], [3, 27, 45], [3, 30, 42], [3, 32, 40], [3, 33, 39], [3, 34, 38], [3, 36, 36], [4, 11, 60], [4, 14, 57], [4, 17, 54], [4, 20, 51], [4, 26, 45], [4, 33, 38], [4, 39, 32], [5, 10, 60], [5, 13, 57], [5, 16, 54], [5, 22, 48], [5, 28, 42], [5, 30, 40], [5, 32, 38], [5, 36, 34], [5, 51, 19], [6, 9, 60], [6, 12, 57], [6, 15, 54], [6, 18, 51], [6, 21, 48], [6, 24, 45], [6, 27, 42], [6, 30, 39], [6, 33, 36], [7, 11, 57], [7, 14, 54], [7, 17, 51], [7, 20, 48], [7, 26, 42], [7, 28, 40], [7, 30, 38], [7, 34, 34], [7, 36, 32], [8, 7, 60], [8, 10, 57], [8, 13, 54], [8, 16, 51], [8, 22, 45], [8, 27, 40], [8, 33, 34], [8, 39, 28], [8, 48, 19], [9, 9, 57], [9, 12, 54], [9, 15, 51], [9, 18, 48], [9, 21, 45], [9, 24, 42], [9, 26, 40], [9, 27, 39], [9, 28, 38], [9, 30, 36], [9, 32, 34], [9, 33, 33], [10, 11, 54], [10, 14, 51], [10, 20, 45], [10, 26, 39], [10, 27, 38], [10, 33, 32], [10, 48, 17], [11, 13, 51], [11, 22, 42], [11, 26, 38], [11, 32, 32], [11, 36, 28], [11, 45, 19], [12, 12, 51], [12, 15, 48], [12, 18, 45], [12, 21, 42], [12, 24, 39], [12, 27, 36], [12, 30, 33], [13, 28, 34], [13, 45, 17], [14, 13, 48], [14, 16, 45], [14, 21, 40], [14, 22, 39], [14, 27, 34], [14, 33, 28], [14, 42, 19], [15, 15, 45], [15, 18, 42], [15, 20, 40], [15, 21, 39], [15, 22, 38], [15, 24, 36], [15, 26, 34], [15, 27, 33], [15, 28, 32], [15, 30, 30], [15, 60], [16, 11, 48], [16, 19, 40], [16, 20, 39], [16, 27, 32], [16, 33, 26], [16, 42, 17], [18, 17, 40], [18, 18, 39], [18, 19, 38], [18, 21, 36], [18, 24, 33], [18, 27, 30], [18, 57], [20, 13, 42], [20, 17, 38], [20, 22, 33], [20, 36, 19], [21, 16, 38], [21, 20, 34], [21, 21, 33], [21, 22, 32], [21, 24, 30], [21, 26, 28], [21, 27, 27], [21, 54], [22, 13, 40], [22, 34, 19], [22, 36, 17], [24, 11, 40], [24, 13, 38], [24, 17, 34], [24, 24, 27], [24, 32, 19], [24, 51], [25, 2, 48], [25, 5, 45], [25, 8, 42], [25, 10, 40], [25, 11, 39], [25, 12, 38], [25, 14, 36], [25, 16, 34], [25, 18, 32], [25, 20, 30], [25, 22, 28], [25, 24, 26], [25, 25, 25], [25, 33, 17], [25, 50], [26, 32, 17], [27, 20, 28], [27, 22, 26], [27, 48], [28, 28, 19], [30, 11, 34], [30, 13, 32], [30, 26, 19], [30, 28, 17], [30, 45], [33, 42], [36, 13, 26], [36, 39], [39, 17, 19], [50, 1, 24], [50, 3, 22], [50, 4, 21], [50, 5, 20], [50, 6, 19], [50, 8, 17], [50, 9, 16], [50, 10, 15], [50, 12, 13], [50, 14, 11], [50, 18, 7]]

194

However, using 48 as a total produces an error message as shown in Figure 2.

Figure 2

I don't understand why sum(c) works when the target is 75 but it doesn't work when the target is 48. However, I found that it would work if I made the total 49 and altered the code from "if target = sum(c)" to "if target - 1 = sum(c)". Weird, right. The program works for 23, 29, 31, 35, 37, 41, 43, 44, 46, 47, 49, 52 and does better as the totals get larger. I can't see any obvious pattern to the misfires.

I imported Numpy and used its sum() function but that didn't work either. The same with Pandas. I put Google's Gemini to work on the problem using this prompt:

This SageMath code works for some numbers e.g. 49 but not for other numbers such as 48. When I use 48 as input, I get the following error message: unsupported operand parent(s) for +: 'Integer Ring' and '<class 'list'>'. The error location is sum(c) but I can't determine what the problem is as I don't understand the error message. 

Gemini was confident that it had identified the problem and even proposed a solution but that proved to be nonsense even though appearing plausible at first glance. I'll have to leave it there and if I do discover the source of the problem I'll discuss it here. Link to Airtable record.