This pattern calls for a size 5 steel hook.

The pattern calls for

Lily Pearl Cotton: 5 balls White, Size 5
Lily Pearl Cotton: 1 ball each Nile Green, Dark Lavender, and 2 balls Salmon Rose in size 3.

I do not know the yardage of this brand of ball in 1945.
There is a smaller square mat and a larger rectangle mat. These mats are worked with some of the rows holding a contrasting “cord” to make the mat thicker and to give it visual appeal. The “cord” is one of the threads listed above.

No gauge or size is given for these.