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This answer to Strange Fruit from Tomato Plant begins:

That looks like a poorly pollinated cucumber (the type of distortion on the fruit is exactly what you'd see if pollination were uneven).

Apparently my understanding of the "birds and the bees" needs to be expanded! I'd always thought that like sperm racing towards a (usually) single ovum in mammals, pollen competed to fertilize a single ovum to make a fruit.

But this comment has caused me to revisit what I'd learned a half-century ago in junior high school, and I've (re-)discovered flowers have multiple ovules.

So, to make a big juicy well-formed cucumber is it necessary for many pollen tubes to fertilize many ovules? How many? Ten? One hundred?

Does each of the cucumber's seeds come from one grain of pollen?

And how can pollination be "uneven" exactly? All the pollen tubes end up fertilizing ovules on one side of the pistil?

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It's not so much the number of ovules that are pollinated, it's WHEN the bees visit that determines successful pollination. This is from the Missouri Botanical Garden:

Weather conditions are key factors in successful pollination. High humidity creates sticky pollen which does not transfer well. Plants in the cucurbit family rely on honeybees for pollination, and honeybees do not fly in cool, cloudy weather. Diseases have dramatically reduced the honeybee population and the ones that are still active remain very susceptible to insecticides.

Cucurbits are monoecious; there are separate male and female blossoms on the same plant. The male flowers tend to open first, followed by the female flowers. It is only when both the male and female flowers are open that pollination can occur. The female flower is open for only one day and is most receptive between the hours of 9 AM and 4 PM. During this time the flower must receive about 15 bee visits for maximum pollination. Unfertilized or poorly fertilized flowers fall from the vine.

So, if it's a humid day or the bees are busy elsewhere, a female flower may not get pollen transferred well from a male flower and/or may not get enough bee visits to completely be pollinated.

Note that, according to the University of Nebraska, the number of bee visits is more of a determinant for adequate pollination than the weather. They also state that only nine visits are needed:

When incomplete pollination occurs, fruit do not develop properly. Because many seeds form within each fruit and each pollen grain is responsible for the development of a single seed, inadequate pollination results in small or misshapen fruit and low yields of marketable fruit. Researchers have found that it takes at least nine honeybee visits per flower to pollinate cucumbers adequately.

The number of visits/grains of pollen transferred to the female flower determines whether a fruit "plumps up" or not because cucumbers (and melons) plumps up around the seeds; missing seeds means that the cucumber won't plump up properly.

There is a way to grow cucumbers that don't require pollination: via parthenocarpic varieties. These varieties do not require pollination to set fruit are are usually grown in greenhouses or under row covers. They also plump up nicely, even without seeds. They can, however, be grown outside as well, but will often set some seed if done so.

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I've read that a single grain of pollen corresponds to a single seed (and each grain of pollen contains two sperm, but both sperm are involved in the fertilization of just one egg/seed, somehow).

Poor pollination could indicate that the fruit contains a lot of inviable seeds, and isn't fully formed, or is undersized as a result. It could mean that they were pollinated by another species of plant that isn't fully compatible (such as a squash, maybe), and so didn't produce viable seeds.

If you're a plant-breeder, it's good to know that each seed in the same fruit can potentially be pollinated by a different father plant (although it's likely that many, if not all, of them have the same father).

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