Ceiling inequality?

Question

I know $\lceil$$\frac{a}{3}$$\rceil$ + $\lceil$$\frac{b}{3}$$\rceil$ $\le$ $\lceil$$\frac{a + b}{3}$$\rceil$ + 1

what do we know about $\lceil$$\frac{a}{3}$$\rceil$ + $\lceil$$\frac{b}{3}$$\rceil$ + $\lceil$$\frac{c}{3}$$\rceil$ + $\lceil$$\frac{d}{3}$$\rceil$?

Answer 1

For $x_i=3q_i+r_i$ we have $\lceil \frac {x_i}3\rceil=q_i+\lceil \frac {r_i}3\rceil$.

For a sum you'll get $\sum\limits_{i=1}^4 \lceil \frac{x_i}{3}\rceil=\sum\limits_{i=1}^4 q_i+\sum\limits_{i=1}^4 \lceil \frac{r_i}{3}\rceil$

While $\bigg\lceil\sum\limits_{i=1}^4 \frac{x_i}{3}\bigg\rceil=\sum\limits_{i=1}^4 q_i+\bigg\lceil\sum\limits_{i=1}^4 \frac{r_i}{3}\bigg\rceil$

So we have to compare $A=\sum\limits_{i=1}^4 \lceil \frac{r_i}{3}\rceil$ with $B=\bigg\lceil\sum\limits_{i=1}^4 \frac{r_i}{3}\bigg\rceil$ for all possible values of the $r_i\in\{0,1,2\}$.

$\begin{array}{|l|cc|c|} \hline r_i & A & B & A-B\\ \hline 0000 & 0 & 0 & 0 \\ 0001 & 1 & 1 & 0 \\ 0002 & 1 & 1 & 0 \\ 0011 & 2 & 1 & 1 \\ 0012 & 2 & 1 & 1 \\ 0022 & 2 & 2 & 0 \\ 0111 & 3 & 1 & 2 \\ 0112 & 3 & 2 & 1 \\ 0122 & 3 & 2 & 1 \\ 0222 & 3 & 2 & 1 \\ 1111 & 4 & 2 & 2 \\ 1112 & 4 & 2 & 2 \\ 1122 & 4 & 2 & 2 \\ 1222 & 4 & 3 & 1 \\ 2222 & 4 & 3 & 1 \\ \hline \end{array}$

Thus the maximum is $2$ and finally we get:

$\displaystyle \big\lceil\frac a3\big\rceil+\big\lceil\frac b3\big\rceil+\big\lceil\frac c3\big\rceil+\big\lceil\frac d3\big\rceil\le\big\lceil\frac {a+b+c+d}3\big\rceil+2$

Answer 2

I don't see the point of those $3$s.

You have $\lceil x\rceil+\lceil y\rceil\le\lceil x+y\rceil+1$. Similarly you have $$\lceil x\rceil+\lceil y\rceil+\lceil z\rceil+\lceil t\rceil \le\lceil x+y+z+t\rceil+3.$$ This is the best you can do. Consider $x$ etc., being just over an integer.