summing data from multiple columns in source data to a single column in the pivot table

not sure if this picture will work
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Here's my challenge: I have a job register spreadsheet showing all the projects we are currently working on - each project is a row of data.
For each project I have listed everyone working on it with the number of days they worked, so I have three columns for names and three for number of days: (associate 1 name, associate 1 days, associate 2 name, associate 2 days, associate 3 name, associate 3 days).
I'd like to calculate the total number of days worked per associate and since they could be listed in associate column 1 for one project and associate column 2 for another project i need to sum days for each associate across these 6 columns.
Any ideas?
When I created a pivot table to do this it took the associate from column 1 and added the days but then summed the total days of any associates listed in associate column 2 and associate column 3 when this associate was listed in column 1.
Essentially I want to add all the days each associate worked on any project all year - whether they were listed as associate 1, 2, or 3.

thanks for your understanding

hi Joseph
thanks for your suggestion. I know I have an unhelpful layout for a pivot here.
I like the idea of an alternative approach and i do have a list of associates - however, I cant make sumif work either.
I would need to set the criteria as 'keep a running total of days every time this name comes up in a list' which i cant figure out how to do. I've tried with a vlookup too but no joy.
Is there a clever way to do this or even a two-step approach?
many many thanks
Juliet