I saw some time ago Carl Sagan asked schoolchildren for proofs that the earth is round. I can no longer find it, but they gave him sound, simple answers. One was watching a ship seem to sink below the horizon as it grows distant. One was photos of Earth from space. One, I believe, was seeing how the sun may reach the bottom of a well in the tropics, but not further north.

I'm trying to find similar answers related to the age of the universe. I'll operationally define "old" as "older than human history." So far:
* We know by parallax distances to various stars within 300 l y, and can tell from their distance, color, and brightness how bright a star of a given color should be. So if we see a further star and measure its color and brightness, we can deduce the distance and thereby how long it took the light to get here. (This is more complicated than I'd like, but oh well.)
* If the universe started X thousand years ago, wouldn't we be seeing more stars appearing every year at the X-thousand l y range? 
* There is a star that exploded? did a nova? and some decades later its light illuminated a nearby nebula. By the time it took to illuminate, we know the distance between star and nebula. Since we know the angle, we can get the length of the other legs of the triangle, which are further than light could travel in human history. (I can't remember the name of the star, alas.)
* Carbon dating.
* The Grand Canyon's sedimentary layers.
* South America and Africa look like they fit together. If they traveled at current speeds of continental drift, they couldn't have reached their current distance in a few thousand years (plus, their animal populations would have mixed when they were touching).

What else?

It's interesting to think of how this works on other questions. Germs, or spontaneous generation of life? proved by canned food. Oxygen, or phlogiston? proved by a sealed-up candle. Quarks? 

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Cepheid variable stars have periodically varying luminosity ( energy of photons released/s), where the period is a function of luminosity. We know observable brightness as a function of distance from a star and its luminosity. By checking if a star varies in brightness, recording its period and measuring its luminosity, we can estimate its distance from us. 

This is in fact part of how we bootstrapped our map of the universe, by using cepheid variable stars as a sort of standard candle. It lets us figure out nearby galactic distances, and hence estimate the universe is >milllions of years old.

In fact, if I recall rightly, they're what allowed to figure out how far away type 1a supernovae were and hence realise they have similair peak luminosities, and hence can be used to figure out distances. 

This isn't really simple though, is it?

Uh, what else. I guess looking at how far away the moon is from the earth + the fact that the earth and moon crashed into each other a long time + its velocity moving away from us could let us guess how long ago they crashed into one another? The maths is not hard, but the physical model is a bit complex. 

I almost want to say something like "simple gravitational models of galaxy fomration imply the universe must be X millions of years old" but I don't know much about galaxy formation, nor do I know how dark matter complicates things. 

Oh, I guess the relative concentrations of uranium and its decay products in the earths crust could be used to estimate the age of the universe + the fact they form in supernovae. 

Uh, the existence of large black holes implies a weak upperbound on the age of the universe, given black hole evaporation rates. 

A fun inverse of this exercise is to go to something like Proofs for a young earth and see how many of them you can counter-argue (and consider how convincing your argument will be to someone with a low level of background knowledge).

With that in mind, I'm not really happy with any of the provided proofs for the age of the universe. While there are a bunch of accessible and intuitively-plausible arguments for getting the age of the earth to at least several million years, determining the age of the universe seems to depend on a bunch of complicated estimates and intermediate steps that are easy to get wrong.

I will definitely check out the "proofs for young earth" thing. A related issue is patching a problem: SA and Africa look like they fit together, and at the current rate of drift they haven't had time to separate in 10K years (haven't checked this, but surely it's right), so maybe they separated 6K years back in a single day. If C14 is really low in things we think are 10M y old (I'm making this up but it fits), maybe they're a few thou years old and a few thou years ago there was very little C14 around.

"SA and Africa look like they fit together" is a good example, because at first glance it looks just a dumb coincidence and not any kind of solid evidence. Indeed, it's partly for that reason that the theory of continental drift was rejected for a long time; you needed a bunch of other lines of evidence to come together before continental drift really looked like a solid theory.

So using the continental drift argument requires you to not just demonstrate that the pieces fit, but include all of the other stuff that holds up the theory and then use that to argue for the age of the earth.

Unfortunately I don't know of any other evidences for the age of the earth or universe that have shorter argument chains. It's genuinely hard! (And partly for that reason I wouldn't be too surprised if new evidence caused us to revise our estimates for the age of the universe by a factor of two in either direction.)

Carbon dating

You're gesturing in the right direction, but if it's the age of the universe you're looking for, you really want something like uranium-lead dating instead, which is routinely used to date rocks up to 4.5 billion years old with precision in the ~1% range. Carbon dating can't reliably measure dates more than ~50,000 years ago except in special circumstances, since the half-life of 14C is 5,730 years. 

Quarks? 

If you fire electrons at protons hard enough, the electrons scatter as if the proton has parts. (For more details, see the 1990 Nobel Prize in Physics.) 

Is this an experiment you can perform yourself, or are you just guessing the teacher's answer?

You can do it yourself, if you can produce a lightning bolt's worth of voltage, but in a way sufficiently sustained and controlled that you don't lose the electrons on their way to the protons. In practice, this has required nation-sized science budgets and facilities the size of a small airport... 

Quarks just aren't something that shows up until you're in a serious era of Big Science subatomic experiment. As late as 1950, one could think that proton, neutron, electron and neutrino are all fundamental and account for all matter (the muon had already been detected, but they thought it was something else). 

It was only when "strange" particles began to appear as tracks produced by cosmic rays, that there was serious evidence of something more, and for some time, it looked like strangeness might be just another basic property (quantum number), like spin or charge. It took the scattering experiments to show that protons actually have parts. 

Glacial ice preserves a record of seasonal temperature variations as each yearly layer of snow accumulates and then partially erodes. The accumulating ice also preserves particles from the atmosphere such as pollen.  Ice cores extracted so far from Greenland extend at least 100,000 years and from Antarctica more than 800,000.

Your last example brings the molecular clock to mind. Maybe like here, with a common ancestor of all current life 4.5 billion years ago.

General relativity seems like a little bit too strong a premise to me