I'm wanting to grow potatoes in a garden bed. Unfortunately, suitable wood is relatively expensive (~$2 a foot for 1x12 cedar) and space is scarce. Would this be too small for half a dozen seed potatoes or so?
My potato bag is one square foot in diameter, and just under 2 foot high. You have 16 square feet in your bed so you could potentially plant one seed potato ( or part of one with at least 3 eyes ) per square foot. One foot deep is about the minimum you can go.
Assuming you have good drainage, and you're using a well fertilized soil, you can plant each about 2 inches off the bottom of the box ( above the ground I'm presuming ), and just cover it to the top. Although some sites advise you to furrow and keep hilling, others say the potato has enough vigor to grow right through those other 10 inches without the need to keep hilling them.
Hilling is where you continue to put dirt around the growing potato plant. Farmers do it to prevent sunlight hitting any new potatoes turning them green, and not to try and get more potatoes. A green potato is toxic throughout the whole potato due to the formation of solanine. The green part is just formation of chlorophyll which is not toxic.
You also want to plant them near the bottom since potatoes grow from modified stem structures so tend not to appear below where the seed potato is placed.
BTW, if you want your cedar to last longer, or, want to use a cheaper wood, you can try preserving it using the shou-sufi-ban method of charring the wood. This protects it against rot, and termites. But I presume you will need a blow torch.
One issue with potatoes is depth: For a good harvest the emerging sprouts are covered with soil to encourage more root mass and more tubers.
In "standard" gardening, this is acchieved by "hills" of soil that are built over the plants, container gardeners start with a shallow layer and fill up as the plants grow.
For your 1ft deep bed, you could think about making it a bit higher (if your budget and garden design allow it), leave it as it is or mound the soil a bit higher than the planks, obviously risking some loss in heavy rain. If your main focus is not maximizing the yield, just use your bed as is.
Other alternatives are bags and similar for potatoes, following the same principle, but you can even roll the rim up and down depending on the required hight at any given time. Some bags are almost 3 ft deep, with good care and nutrition, they result in a lot of potatoes per bag...