The Pencil Room

Still open

• Why do we dream?

  Scientists can see which parts of the brain are busy while you dream. Nobody yet knows what dreams are for.

  Still open, says Mota-Rolim, Holzinger, Nadorff & De Gennaro, "In the Arms of Morpheus", Brain Sciences (2024): we still do not know why we dream

• How many kinds of living things are there?

  One careful guess is about 8.7 million kinds of plants, animals and fungi, and most of them have never been named. Nobody has counted.

  Still open, says Mora, Tittensor, Adl, Simpson & Worm, "How Many Species Are There on Earth and in the Ocean?", PLoS Biology (2011): some 86% of existing species on Earth ... still await description

• How do birds find their way across the world?

  Birds can feel the Earth's magnetism. Scientists think a tiny part of their eyes helps, but nobody has found the sensor for sure.

  Still open, says Institute for Bird Research (Vogelwarte Helgoland), cryptochrome project: Identifying the primary magnetosensor remains a fundamental open question.

• Is there life on other planets?

  Astronomers have found thousands of planets around other stars. So far the only life anyone knows about is here.

  Still open, says NASA Science, "The Search for Life": But so far, we have no evidence of life beyond Earth.

• What is most of the stuff in space made of?

  Most of the matter in the universe pulls on stars and galaxies but cannot be seen. It is called dark matter, and nobody knows what it is.

  Still open, says NASA Science, "Dark Matter": scientists have not yet found evidence which would confirm any candidate

• Does every number fall to 1?

  Pick a number. If it is even, halve it. If it is odd, triple it and add 1. Keep going. Every number anyone has tried ends at 1, but nobody has proved they all do.

  Still open, says Kevin Hartnett, "Mathematician Proves Huge Result on 'Dangerous' Problem", Quanta Magazine (2019): Tao's result is not a full proof of the conjecture

• Do the twin primes ever stop?

  Twin primes are prime numbers two apart, like 11 and 13. They keep turning up, but nobody knows if they go on forever.

  Still open, says Wolfram MathWorld, "Twin Prime Conjecture": It is not known if there are infinitely many such primes

Carry one with you

Pick one question to wonder about this week. It stays on this device only.

Inked: answered at last

These were open questions once. Someone answered them, so they are in ink.

• Can every map be coloured with four colours? inked 1976

  People wondered for over a hundred years. In 1976 two mathematicians proved yes, with help from a computer: four colours are always enough so that no two neighbours match.

  Settled in Appel & Haken, "Every planar map is four colorable", Illinois Journal of Mathematics (1977)

• What is the tightest way to stack oranges? inked 2017

  Johannes Kepler guessed in 1611 that the shop stack is best. A proof was finished in 1998, and a computer checked every step of it in 2017.

  Settled in Hales et al., "A Formal Proof of the Kepler Conjecture", Forum of Mathematics, Pi (2017)