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**Hint:**We know that in Newman's projection, any particular carbon-carbon bond is viewed along its longitudinal axis. The carbon having a lower number is placed at the front and denoted as a point. While the other carbon is placed at the back and denoted as a circle. The most stable conformer is the one having two bulky groups placed at opposite ends and having minimum steric strain.

**Complete answer:**

Let’s view the $3,3-$dimethylhexane structure in a simple form before drawing its Newman projection.

As we have to find for the $C_3$ and ${{C}_{4}}$ we see that there are two methyl groups and one ethyl group on ${{C}_{3}}$ and on ${{C}_{4}}$ two hydrogen and one ethyl group.

Above diagram is a representation of Newman first staggered projection. Now for the second and third staggered Newman projection we rotate the atom by ${{120}^{\circ }}$,we get

The above projection is less stable than the first projection made before rotation. In the above projection we have gauche interactions between two bulky ethyl groups and a methyl group. The repulsion between two methyl groups and one ethyl group being a little less unstable than the repulsion the other two staggered conformers have in the second and third projection. The projection before rotation has the most stable Newman’s projection. It also has both ethyl groups in anti-position.

**Note:**

Remember that all the conformers are not very stable at room temperature and are rapidly interconverting to each other. The most stable conformer is said to be the one having bulkier groups away from each other due to the less steric hindrance between such groups.

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