Table nesting with columnar widths based on uniform parent width, not uniform child width

Question

I think I am using the wrong search queries to say this, but I'm trying to make a table that contains cells whose widths are determined by their parents' uniformity, not whose parents' widths are determined by their childrens' uniformity.

My example is I'd like to make my table look like this:

As you can see, the titlebars for col1 == col2 == col3 in width.

Child cells x1 == x2 == x3, and y1 == z1, but x1 != y1.

However, the code I have below is producing even child cells, and differently sized parent cells:

\begin{table}
\setlength\tabcolsep{0pt} 
\begin{tabu} to \textwidth { | *{5}{X[c]|} | *{3}{X[c]|} | *{3}{X[c]|} }
    \toprule
    \multicolumn{5}{c}{col1} & \multicolumn{3}{c}{col2} & \multicolumn{3}{c}{col3} \\
    %% \cmidrule{1-5} \cmidrule{6-8} \cmidrule{9-11} \\
    x1 & x2 & x3 & x4 & x5 & y1 & y2 & y3 & z1 & z2 & z3 \\
    \midrule
    1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 \\
    \bottomrule
\end{tabu}
\caption{mess.}
\label{tab:mess}

\end{table}

As you can see:

What should I do differently?

Answer 1

I think you can achieve your formatting objective by switching to a tabularx environment and using the environment's ability to specify relative column widths.

If the first main column has 5 equal-width sub-columns, and if the second and third main columns have (essentially) no sub-columns, the relative width of each of the first 5 subcolumns is given by 7/15=0.4667. Why 7 and 15? Because (a) there are effectively 7 columns and (b) one can think of the entire tabular-like environment taking up 5*3=15 columns. The relative widths of the second and third main columns are given by 5*0.46667=2.3333. Note that 5*0.4667+2*2.3333=7, the number of columns of type X. (Aside: The math is much simplified by the fact that you're executing \setlength\tabcolsep{0pt}!)

I'm assuming that there are, effectively, no subcolumns for columns 2 and 3. I'm making this assumption because you've indicated that you plan to show equations in these two columns. I believe it's better to center the entire equation within the column, rather than simply center the = symbols.

\documentclass{article}
\usepackage[margin=1in,letterpaper]{geometry} % set suitable page parameters
\usepackage{tabularx,booktabs}
% define a flexible-width centered version of "X" column type:
\newcolumntype{C}[1]{>{\centering\arraybackslash\hsize=#1\hsize}X}
\begin{document}
\begin{table}
\setlength\tabcolsep{0pt}
% Total number of C-type columns: 7
% Number of (implicit) columns of equal width = 3*5 = 15
% Relative width of 5 subcolumns of col. 1: 7/15=0.4667
% Note: 5*0.4667+2*2.3333 = 7 = # of C-type columns 
\begin{tabularx}{\textwidth}{ *{5}{C{0.4667}} *{2}{C{2.3333}} }
\toprule
\multicolumn{5}{@{}c}{header 1} & header 2 & header 3 \\
\cmidrule{1-5}
x1 & x2 & x3 & x4 & x5 \\
\midrule
1 & 2 & 3 & 4 & 5 & yyyyy & zzzzz \\
\addlinespace
a & b & c & d & e &
$\displaystyle v=\int_{-\infty}^{\infty} e^{-x^2}dx$ & 
$\displaystyle w=\sum_{i=1}^{\infty}\frac{1}{i^2}$ \\
\bottomrule
\end{tabularx}
\end{table}

\end{document}