What are some unsolved problems in math that seem easy at first glance (e.g., the Collatz conjecture)?
1. Here is an unsolved problem that is probably the easiest on this list to state in a formal language whose foundational object is the set:Suppose you have a collection of sets where for every two sets A and B in the collection, A∪B is also in the collection. (This is called the "union-closed" property.) If your collection is finite and non-empty, is it always true that there is some element that is present in at least half of the sets in the collection?If so, the "Union-closed Sets Conjecture" would be true, but this has been unresolved for more than 34 years. It's been verified for collections built from at most 12 elements, which I thought could be improved with a little dose of modern GPU power, but even that is hard.Amazingly, it's not even known if there must exist an element that is present in at least, say, 1% of the sets.2. As a bonus, here is a geometric problem: Is it possible to slice a cake into congruent pieces (more than 1) so that the center of the cake is in the interior of one of the pieces?By "cake" I mean a two dimensional disc, of course. Also, pieces are congruent if their interiors are identical after translation, rotation, and/or reflection.I first saw this problem at Unusual Cake Slicing, which also shows a solution to a related question: Can a cake be sliced into congruent pieces such that the center of the cake doesn't touch some of the pieces? (By "touch" it means that the center is in the closure of the piece.) It's also mentioned at the bottom of page 87 in the 1991 printing of "Unsolved Problems in Geometry" by Croft, Falconer, and Guy.
I Wake up at night to see strange shapes and objects in my room!?
This is strange i know, and I cant seem to find any answers or threads.. On most nights i am terrified of my own room.. i will be in bed.. most of the time.. i wake up during the night to see.. abstract round blob like shapes.. that go float past my wall.. or travel on the floor, roof and over me or see shapes (similar to i guess.. science cells when looking in a microscope..) some BIG and some small and all colours.. the other night i woke up.. to see a floating small-medium long thick stick instect almost looking hovering at the entrance of my door inside my room.. which these disappear and it ventured to my hallway outside my room..one time it was morning.. an after seeing on of these things.. i heard like a clear male grin with a "pstt" right in my ear it was soo real and i was alone in bed.. some night terrified to go to my room just because i know ill see one of these things.. I CANT STAY IN MY ROOM WHEN THIS HAPPENS! I will get serious goose-bumps and start shaking as i lay still.. walkin in my room or in bed sometimes.. PLEASE if there is anyone with a similar story let me know! :) PS: I AM VERY HEALTHY YOUNG LADY, SLEEP IS NORMAL, AND I DONT HAVE ANY STRESS RELATED CAUSES, NOR DO I DRINK OR TAKE ANY DRUGS OF ANY SORTS.. Regards Mina
Physics: Explain Quantization to a Kid using money as the quantized entity..?
Quatization is when things are divisible into steps, or packets. Money is quantized because you can't carry around 1.75 cents in your pocket. The smallest quanta of money you can carry, in the US anyway, is 1 cent, and the closest you can get to 1.75 cents is 2 cents. You can only increase your money by whole numbers because coins and dollars only come in whole numbers. The same is often true of how much energy something can have, or will react to.
If a motor neuron involved in a reflex arc is damaged, which event in that arc is least likely to occur?
The reflex arc goes like this: Receptor > Sensory neuron (afferent) > Interneuron > Motor neuron (efferent) > Effector organ (muscle or glands) So if the motor neuron is damaged, you will receive the stimulus, send the information through the sensory neuron and interneuron, but if the motor neuron is damaged, it won't make it to the muscle to be able to contract. (Answer A.)
What marks a year in a geocentric system?
The key determinate for a year is the due to the relative tilt of the earth to the plane of the ecliptic. Winter days are shorter than summer days. You can do this yourself in your backyard. Put a stake in the ground. Mark the tip of the shadow during the day. On successive days as we approach winter solstice (I'm writing in the northern hemisphere in October) you will see these arcs getting further and further from the post. On Dec 22 they reach a maximum. December 23's arc is very close to the 21st'sYou can do other things that measure the same sort of thing. Sunrise in summer is in the north east. In winter, the south east. On fall equinox it is due east.The boarding school I worked in had a hallway 300 feet long with outside doors at the east and west ends. Those doors had 8 x 30" windows in them. On the equinoxes sunrise would shine through the east door window and hit the west door. One day either way, and it hit the side of the hall instead.Egyptians used Sirius The rising of Sirius relative to sunset foretold when the Nile would flood.