Coding classes in the Vale of Glamorgan

Start with certainty. If 701 birds share 700 holes, at least one hole must hold two, whatever the birds do. Mathematicians call this the pigeonhole principle. The learner codes the general rule: with n birds and 700 holes, some hole holds at least n divided by 700, rounded up. So 1,400 birds could fit exactly two to a hole, but a 1,401st forces a hole to take three. No simulation is needed; the answer is guaranteed.

Now probability. Suppose each bird picks any hole with equal chance; real pigeons are not so random, so this is our invented behaviour. The chance a particular hole is missed by every one of 700 birds is (699/700) to the power 700, which is about 0.368, very close to 1 divided by e. So even with exactly one bird per hole available, about 257 holes stay empty while others fill up. The learner checks the formula by simulating it 2,000 times: the average comes out at 257.2, and the fullest hole usually holds about five birds.

Last, the size check. The record gives an inside diameter of about 5 metres and walls about 4.6 metres high to the start of the dome. The inside wall then has an area near 72 square metres, and 700 holes of 15 by 15 centimetres use under 16 of them, about 22 per cent, which leaves room for stone between. Sixteen rows of about 44 holes would put their centres roughly 36 centimetres apart. So the 700 is believable. The slip to catch: multiply the diameter by two pi instead of pi and the wall doubles to 144 square metres.

Telling a proof from a probability is a clear reason for a Vale of Glamorgan teenager to keep coding in 2026. The longer argument is in why teenagers should still learn to code in 2026.