Permanently keeping the 1MB (anti-spam) restriction is a great idea ...

[zimmah]

The blockchain works by supply and demand, and supply is made by the miners themselves, not by some dumb hardcoded limit.

You assume that miner can effectively control supply. This in absence of a block size limit is only the case if they are building cartels, that artificially limit the supply.

Is that you really prefer instead of an algorithmic decision?
Remember Bitcoin's promise is to operate without the need of cartels and authorities.

oh but you can.

the miners decide which transactions are valid and which are not.

you can make your own rules about which fees to accept and which fees not to accept. You can even ban individual transactions or addresses if you want.

I'm pretty sure there's also a way to make your own limits to the size of the block you want to solve, as long as it's not bigger than the hard-coded limit.

As long as your block is considered "valid" by the rest of the network you decide the rules about which transactions you accept and which you dont accept.

If you have sufficient power as a miner, you can have a pretty big influence on the fees. And any miner who's not stupid will ask for at least a small fee. Or like make something like a 250KB limit for 'free transactions' and reserve the rest of the block for transactions with fee, the more fee the more priority, something like that.

be creative...

just because there's the POSSIBILITY of 20MB doesnt mean you HAVE TO use it.

There's not even a need for cartels, but each miner/pool will set their own limits based on some calculations.

Do i really need to do all your thinking for you?

Calculations would be roughly based on

• how much electricity does mining cost for me?
• how much $/kWh in my area?
• how much transactions are in an average block lately?
• how many blocks do i solve (or more likely does your pool solve) per hour
• how much of a % do i influence the pool?
• how much is the block reward if any? (and how much does it cover the cost of mining?)
• how long does this mining equipment stay competive
• how much did my equipment cost and how much will i need to profit to invest in new one
• overhead costs

weight all those factors against each other and you can make a rough estimate on how much fees you should ask for an average transaction.

now let's suppose we're far into the future and we rely mostly on fees as opposed to block rewards.

lets also assume there's (for simplicity sake) 8 large pools controlling a majority of the mining. (not to say this will happen but just to keep the calculations easy).

Pool A asks for a fee of 1 bitcoin minimum. and controls 25% of the network
Pool B asks for a fee of 0.001 bitcoin minimum and controls 5% of the network
Pool C asks for a fee of 0.005 bitcoin and controls 10% of the network
Pool D asks for a fee of 0.003 bitcoin and controls 10% of the network
pool E asks for a fee of 0.5 bitcoin and controls 15% of the network
pool F asks for a fee of 1 satoshi and controls 10% of the network
Pool G asks for a fee of 0.01 bitcoin and controls 15% of the network
Pool H asks for a fee of 0.025 bitcoin and remains the remaining 10%

now if i would want to sent a transaction and be absolutely sure it gets confirmed 6 times within the next 6 blocks (which should take about an hour) I should add a fee of 1 bitcoin.

However, if i'm satisfied with a 75% chance of waiting for a block i just add a fee of 0.5 instead.

If i am satisfied with a chance of 60% to have to wait a block, i'd add a fee of 0.025 and so on.

Of course as the user I don't know exactly what the miners ask in fees, so it's a bit of trial and error. But most users will add a fee that proved to work for them in the past.

Many large transactions or transactions that NEED to be processed fast will likely have a higher fee to ENSURE a fast transaction (some miners only accept large fees) smaller transactions (which will be more plentiful) will add lower fees, risk waiting a little bit longer.

The miners who accept smaller fees will process more transactions and the sheer volume of transactions may actually add up. But the few big transactions that the other pools sometimes pick up may also bring in some large chunks of fees from time to time.

There's no need for a hard-coded limit, fees will come in anyway, and you can make your own rules.

[grau]

^ Did you read the entire post? The OP fully addressed the effect on fees:

He neglects that there is no reason to pay fees, if there is no limit on supply.

just because there's the POSSIBILITY of 20MB doesnt mean you HAVE TO use it.

Since there is no marginal cost in including a transaction to the current block, a rational miner will always include a transaction with a non zero fee,
before it is included by any of its competitors.

Therefore a lower bound on fee will not work without a cartel or without a competition for space.

I prefer algorithms over cartels.

[ABitNut]

You must be able to broadcast that huge block to most of the nodes in 10 minutes. I don't see the latest research regarding this area, but there is a paper from 2013


http://www.tik.ee.ethz.ch/file/49318d3f56c1d525aabf7fda78b23fc0/P2P2013_041.pdf

Based on this research, it took 0.25 seconds for each KB transaction to reach 90% of network. In another word, a 1MB block will take 256 seconds to broadcast to majority of nodes and that is 4 minutes

When block size reach 10MB, you will have a broadcast time of 40 minutes, means before your block reach the far end of the network, those nodes have already digged out 3 extra blocks thus your block is always orphaned by them. And the whole network will have disagreement about which segment have the longest chain, thus fork into different chains

Gavin's proposal is to let mining pools and farms connect to high speed nodes on internet backbone. That is reasonable, since the propagation time is only meaningful for miners, your transaction will be picked up by the mining nodes closest to you and if those mining nodes have enough bandwidth, they can keep up with the speed. But anyway, how much bandwidth is really needed to broadcast 10MB message in a couple of minutes between hundreds of high speed nodes need to be tested. And this is the risk that someone worried about the centralization of mining nodes: Only those who have ultra high speed internet connection can act as nodes (I'm afraid that chinese farms will be dropped out since their connection to the outside world is extremely slow, they will just fork to their own chain inside mainland china)

I don't know how you come to those assumptions based on that research.

the block message may be very large — up to 500kB at the time of writing.

The median time until a node receives a block is 6.5 seconds whereas the mean is at 12.6 seconds.

For blocks, whose size is larger than 20kB, each kilobyte in size costs an additional 80ms delay until a majority knows about the block.

The do not mention the average size of blocks they measured. Let's assume all their blocks were 0KB. 12.6 seconds for that. Add 80 ms per addicition KB.... 80ms * 1024 * 20 is about 27.3 minutes. Add the original 12.6 seconds... Roughly 28 minutes for 20MB.

Of course, 28 minutes is still long. That is based on 2013 data. I assume the nodes now will have improved their verification speed and have more bandwidth. New measurements could / should be made to verify that propagation speed will not become an issue.

[grau]

Since there is no marginal cost in including a transaction to the current block,

let me be more precise:
There is a marginal cost implied by block propagation speed being proportional to size and propagation being proportional to orphan rate. There is also a computation cost of updating the merkle tree and updating miner with it. These marginal costs are today however magnitudes below the lowest non-zero fees paid.

[zimmah]

^ Did you read the entire post? The OP fully addressed the effect on fees:

He neglects that there is no reason to pay fees, if there is no limit on supply.

just because there's the POSSIBILITY of 20MB doesnt mean you HAVE TO use it.

Since there is no marginal cost in including a transaction to the current block, a rational miner will always include a transaction with a non zero fee,
before it is included by any of its competitors.

Therefore a lower bound on fee will not work without a cartel or without a competition for space.

I prefer algorithms over cartels.

then why does a transaction with a 1 satoshi fee take longer to process even though we are not at our limit yet?

explain me this.

[grau]

then why does a transaction with a 1 satoshi fee take longer to process even though we are not at our limit yet?

explain me this.

Because majority of miner nowdays use the bitoin core as published by the devs and that compiles blocks following a mix of commercial and alturistic rules.

Majority of miner do not care about this since inflation provides three-four magnitudes higher income than fees at the moment. They will. And this discussion is about the future and the limits.

[zebedee]

I am not the only one totally annoyed by how bitcoin behaves in an effort to keep the dominant position and be the 'one coin for all' and with this hurts the alt-industry. I think at this point it becomes inevitable to start using more than one chain and stop looking at btc as the only coin worth bothering with.

I see, so rather than doubling the size of the bitcoin chain you'd rather have two chains half the size?  How is that a win?

[turvarya]

Since I so often see the argument about, what is good for the miners:
Is there any statement from the large mining pools about this?

[zimmah]

then why does a transaction with a 1 satoshi fee take longer to process even though we are not at our limit yet?

explain me this.

Because majority of miner nowdays use the bitoin core as published by the devs and that compiles blocks following a mix of commercial and alturistic rules.

Majority of miner do not care about this since inflation provides three-four magnitudes higher income than fees at the moment. They will. And this discussion is about the future and the limits.

well, in the future miners will either adapt and make their own rules, or leave the mining industry because they are too broke to continue. And smarter miners will take over.

judging by your post, you'll be one of the broke ones, because you're too lazy or stupid to make your own rules.

it's the miners that make the rules, not the protocol.

but the protocol should not be a hard limit, because that would just destroy bitcoin.

I'm not going to waste my time thinking of more examples to point this out, i have better things to do.

[grau]

judging by your post, you'll be one of the broke ones, because you're too lazy or stupid to make your own rules.

it's the miners that make the rules, not the protocol.

but the protocol should not be a hard limit, because that would just destroy bitcoin.

I'm not going to waste my time thinking of more examples to point this out, i have better things to do.

You know much less about Bitcoin that you think. I hope you have better things to do.

Miner do not make the rules. They have to obey the rules if in minority or can narrow existing rules if organizing a majority cartel.
The later would be needed to impose a non-zero fee in absence of a size limit in the protocol.

[homo homini lupus]

I am not the only one totally annoyed by how bitcoin behaves in an effort to keep the dominant position and be the 'one coin for all' and with this hurts the alt-industry. I think at this point it becomes inevitable to start using more than one chain and stop looking at btc as the only coin worth bothering with.

I see, so rather than doubling the size of the bitcoin chain you'd rather have two chains half the size?  How is that a win?

It IS in fact a win - coins are easy to convert in many different ways within splitseconds. How is it a loose? From an economic standpoint it's not a loose. Say the coffeeshop accepts 'Bitcoin' but i do use xyz-coin. The conversion is a mere formal act. I can pay the coffeeshop in xyz-coin and the shopowner gets the coin he wants (gavincoin or whatever). Crypto is easy enough to convert for this to have no economic impact. All it takes is an exchange-app, and i think it could already be developed or even exist already (would need to check)

With multiple smaller chains the enduser doesn't loose souvereignity to too high hardware requirements.

You could in fact even merge mine them.

The individual user would not be required to store Terrabytes of data and have extreme bandwidth like he would be with Gavincoin.

How is requiring everyone to potentially store terrabytes a win?

---
This scenario is what will happen most likely anyway once running Gavincoin becomes unaffordable for the enduser.

[Nancarrow]

The OP gives a valid technical argument for raising block size limit, but is neglecting a financial argument against it.

The miners' income has to be greater than the cost of their work. Miners' income is inflation now, but is expected to be replaced by fees,
since inflation halves every four years. Purchasing power of new coins might be sustained for a while but must converge to zero in the limit.

Transaction fees exist only because there is a competition for block space. Eliminating that competition eliminates the fees and with that mining.

Therefore block space has to become and remain a scarce asset.

Your second and third paragraph contradict each other.

Transaction fees don't ONLY exist because there is a competition for block space. They ALSO exist to pay the miners to secure the network, as you clearly understood before you implicitly denied it. Fees are not an either/or thing. It absolutely ISN'T a case of lifting block limit->eliminates fees->eliminates mining.

We have instead a *feedback* process. LOWER fees (not ZERO fees) means LESS mining (not NO mining) which in turn means LONGER confirmation times (not COMPLETE COLLAPSE) which leads to MORE FEES which leads to mining power switching back on. It's what engineers call a negative feedback loop, designed to keep the hashing rate broadly stable, or at least oscillating within a fairly narrow range.

People really must stop thinking of all the causes and effects in the world as being ON/OFF switches. They aren't. They're analogue dials.

[If you're on board with the idea of bitcoin, you've probably had to deal with people saying a deflationary money supply can't work because NOBODY would ever spend ANY MONEY AT ALL. Same problem. "Less" is not the same as "none". Especially when "Less X" induces "Less Y" which induces "More X".]
 

[zimmah]

judging by your post, you'll be one of the broke ones, because you're too lazy or stupid to make your own rules.

it's the miners that make the rules, not the protocol.

but the protocol should not be a hard limit, because that would just destroy bitcoin.

I'm not going to waste my time thinking of more examples to point this out, i have better things to do.

You know much less about Bitcoin that you think. I hope you have better things to do.

Miner do not make the rules. They have to obey the rules if in minority or can narrow existing rules if organizing a majority cartel.
The later would be needed to impose a non-zero fee in absence of a size limit in the protocol.

you just assume all miners will allow a low fee, but that simply won't be the case.

because it's not sustainable.

for the same reason taxis do not offer free rides, and shops dont offer free products.

like i explained before, you need to calculate a lot of factors, and then decide a cut-off price. And all the miners who dont do that once the fees start to matter more than the block reward will go bankrupt and will be forced to stop.

just because the transaction itself doesnt cost anything (or barely anything) doesnt mean it's free.

Everyone understands that at some point transaction won't be free, because miners need compensation for their power and other costs.

The thing you refuse to understand is that it's better to spread that cost over 1 million transactions per 10 minutes than on 1620 transactions (which, based on empirical evidence is about the limit of the current blocksize).

Because if you spread the fee over just 1620 transactions, the fee would be ridiculously high and would never be able to support the network, but if you spread it over a million transactions it will be very cheap. But you wont be able to spread the cost over a million transactions because it simply would not fit in 1 MB.

You were talking about the future, so think about the future. If bitcoin still has only 2 transactions per second on average by the time block rewards are insignificant, bitcoin has failed, and this whole discussion is pointless.

Are you really this stupid or are you just trolling?

by the way if miners dont make the rules, who do?

[zimmah]

I am not the only one totally annoyed by how bitcoin behaves in an effort to keep the dominant position and be the 'one coin for all' and with this hurts the alt-industry. I think at this point it becomes inevitable to start using more than one chain and stop looking at btc as the only coin worth bothering with.

I see, so rather than doubling the size of the bitcoin chain you'd rather have two chains half the size?  How is that a win?

It IS in fact a win - coins are easy to convert in many different ways within splitseconds. How is it a loose? From an economic standpoint it's not a loose. Say the coffeeshop accepts 'Bitcoin' but i do use xyz-coin. The conversion is a mere formal act. I can pay the coffeeshop in xyz-coin and the shopowner gets the coin he wants (gavincoin or whatever). Crypto is easy enough to convert for this to have no economic impact. All it takes is an exchange-app, and i think it could already be developed or even exist already (would need to check)

With multiple smaller chains the enduser doesn't loose souvereignity to too high hardware requirements.

You could in fact even merge mine them.

The individual user would not be required to store Terrabytes of data and have extreme bandwidth like he would be with Gavincoin.

How is requiring everyone to potentially store terrabytes a win?

---
This scenario is what will happen most likely anyway once running Gavincoin becomes unaffordable for the enduser.

you really dont get it do you?

even if we ignore the legion of premined scamcoins, we still have the following issue:

Suppose we use 4 different altcoins

Acoin has a 10 GB blockchain in this example
Bcoin has a 5 GB blockchain
Ccoin has a 20 GB blockchain
Dcoin has a 15 GB blockchain

now how exactly is that better than having 1 blockchain of 10+5+20+15=50 GB?

If you are going to store them all anyway, it doesnt matter if you have 1 blockchain of 50 GB or many smaller ones adding up to 50.

In fact, having multiple blockchains means you have a lot of extra redundant information and applications running, it's less convenient, less secure, etc. It only brings more problems and it doesnt solve anything.

Why am i even explaining this....

[amincd]

^ Did you read the entire post? The OP fully addressed the effect on fees:

He neglects that there is no reason to pay fees, if there is no limit on supply.

So now that you've moved the goalposts (first it was: "he doesn't address fees"), I need to correct you again. The OP never advocated for there being no limit:

The problem isn't a limit in general but that 1MB is so low that under any meaningful adoption scenario it will push all individual users off the blockchain to rely on trusted third parties.

A limit with thousands of tps will undoubtedly produce more fees for miners than a limit capping the network at 3 tps.

[Nancarrow]

Oh another thing: re the concern about a huge blockchain size...

Increasing the block limit isn't the only thing Gavin and co have been giving a lot of thought to. They've also understood the (complementary) problem of blockchain bloat. I'm given to understand that several blockchain pruning schemes are out there. Note these are NOT the same idea as 'lightweight' or SPV clients. They're genuine info compression schemes. I haven't followed for a while, but one scheme I heard basically suggested that *on disk* the blockchain information up to the 'current block #X' should essentially be split into three parts:

1) A chain of block headers only up to block #X-N, where N might be a couple of thousand.
2) A database of UTXOs for all the blocks up to #X-N
3) A complete sequence of the last N blocks

The header-only chain provides the proof of the next block's valid membership of the chain. It grows linearly for ever more, at a rate of ~(640*6*24*365)/(1024^2) = 32 MB a year. Which is peanuts.
The UTXO database grows more unpredictably, but I guess broadly proportionally to global bitcoin usage. It's there so the node knows who's got what and can verify new transactions are valid.
The uncompressed sequence of the last N blocks, where N might vary by individual preference, remains fixed length for a fixed blocksize. The node keeps the last N rolling blocks uncompressed in case there's a fork.
Even if the blocksize grows, so that this tail of blocks grows, it's unlikely to gum up disk space. If it does, the node could choose to reduce N (I pick N=2000 as rather overly cautious).

Anyway TLDR: disk space is NOT a problem. Can't comment much on network propagation issues, except I'm led to believe these can be addressed by an approach such as 'send the header first, then the block info separately'.

[homo homini lupus]

I am not the only one totally annoyed by how bitcoin behaves in an effort to keep the dominant position and be the 'one coin for all' and with this hurts the alt-industry. I think at this point it becomes inevitable to start using more than one chain and stop looking at btc as the only coin worth bothering with.

I see, so rather than doubling the size of the bitcoin chain you'd rather have two chains half the size?  How is that a win?

It IS in fact a win - coins are easy to convert in many different ways within splitseconds. How is it a loose? From an economic standpoint it's not a loose. Say the coffeeshop accepts 'Bitcoin' but i do use xyz-coin. The conversion is a mere formal act. I can pay the coffeeshop in xyz-coin and the shopowner gets the coin he wants (gavincoin or whatever). Crypto is easy enough to convert for this to have no economic impact. All it takes is an exchange-app, and i think it could already be developed or even exist already (would need to check)

With multiple smaller chains the enduser doesn't loose souvereignity to too high hardware requirements.

You could in fact even merge mine them.

The individual user would not be required to store Terrabytes of data and have extreme bandwidth like he would be with Gavincoin.

How is requiring everyone to potentially store terrabytes a win?

---
This scenario is what will happen most likely anyway once running Gavincoin becomes unaffordable for the enduser.

you really dont get it do you?

even if we ignore the legion of premined scamcoins, we still have the following issue:

Suppose we use 4 different altcoins

Acoin has a 10 GB blockchain in this example
Bcoin has a 5 GB blockchain
Ccoin has a 20 GB blockchain
Dcoin has a 15 GB blockchain

now how exactly is that better than having 1 blockchain of 10+5+20+15=50 GB?

If you are going to store them all anyway, it doesnt matter if you have 1 blockchain of 50 GB or many smaller ones adding up to 50.

In fact, having multiple blockchains means you have a lot of extra redundant information and applications running, it's less convenient, less secure, etc. It only brings more problems and it doesnt solve anything.

Why am i even explaining this....

How do premined scamcoins even matter? They don't.


Let's go out in the future 5 years from now:

Gavincoin: 1.5 terrabyte storage requirements
Mpcoin: 70GB
Acoin: 10gb
Bcoin: 20gb
Ccoin: 5gb
Dcoin: 1gb


you don't store them all - especially not gavincoin. You store two or three maximum. Those that are most according to your usecase of crypto.
So your assumption everyone would store all chains is wrong of course. Different people use different coins for different purposes and will use different chains according to their preferences.

a,b,c,d,e coin can still be merged mined with the strongest network (which one that will be remains to be seen) - so a,b,c,d coin are almost as secure as the network they merge mine against.

Almost nobody will use a coin that requires such insane hardware as Gavincoin will.
Once this is understood there's going to be a lot developement towards efficency - Gavincoin is a proposal for a waste of systemrecources and therefore it won't be able to compete with more efficient coins in the long run.

If in any case i would be required to use Gavincoin i can still convert my a,b,c,d,e coin in splitseconds to it.

[homo homini lupus]

Not the coin that's the biggest bully and has the most vocal support wins but the coin that's most efficient. (not even network effect and merchant adoption are as important as efficency in the long run)

So fork it all up, fools. Do it right now!

[Stephen Gornick]

On edit: fixed some typos.

Possibly one more ...
"Have you ever thought about the fact that you send a bank wire yourself."
Did you mean to instead write "that you can't send a bank wire yourself."?

[Stephen Gornick]

If I lose direct access to the blockchain then I am forced to hand over all my wealth (not just enough funds to cover trivial purchases) to a third party.

That's probably the best argument I've seen.  I think of how today I have many paper wallets, mobile wallets, and hosted (shared) E-Wallet services (i.e., custodial services) where I had a balance of a few millibits or less in them and won't (or can't) withdraw because the Bitcoin network transaction fee is prohibitive for that small amount of bitcoin.    Now as fees rise due to scarcity of empty space in blocks then I'll essentially have permanently abandoned those wallets with the small values (i.e., those funds become economically unspendable, ... worthless!).

I suppose for the paper wallets and mobile wallets where I have the private key I could combine them into a single transaction without incurring a higher fee, but that's possible only today because there's still some low-cost space left and the fee required doesn't rise linearly when the size of the transaction rises.   Without a fork then eventually we are essentially paying a fee for each additionally byte and the resul will likely be that millions of UTXOs with small amounts (e.g., sub-millibit -- less than 0.001 bitcoins) will become worthless.   Hey now,  that's some real money we're talking about discarding!  [Anyone care to crunch the numbers -- what is the sum total of bitcoins that exists in UTXOs under one millibit?)  

It makes me think back ... First they came for the dust, and I didn't speak out because I didn't play SatoshiDICE.    When this day comes (when the 1MB cap is reached, and no-hard fork), maybe a millibit becomes the new dust!   And then let's say the fee required rises even more, maybe UTXOs under 0.01 even become economically unspendable.    That becomes a real problem for most of us!

So what to do?
I'ld like a Bitcoin that can scale and grow with transaction demand.  I'ld also like my car to get 150 MPG.  The latter can't happen simply because a gallon of gasoline doesn't have the energy needed for an internal combustion engine to propel today's car for 150 miles.  Now I'm not entirely convinced the former can happen either.  That's because a hard fork that doesn't have the consent of the Economic Majority ( http://en.bitcoin.it/wiki/Economic_majority ) will fail.

Let's go through the first hours of the hard-fork.  Let's say it happens at block 400,000 (a lttle over a year from now).  Everything was in place -- miners with the right nVersion indicating consent was well above the threshold (e.g., 80% of last 1,000 blocks had nVersion=4).     But ..., I'm not willing to believe that exchanges, merchants, merchant processors, etc. are going to themselves take on the full risk of double spending that would occur if the hard fork eventually fails (maybe a day or three later even).   The only way to prevent the double spending that would result would be to require that a transaction confirms on the block chains on both sides of the fork.  So these entities are going to watch both sides. Well, a transaction that has any taint from a coin generated on the side with the larger block size rule change (I hate calling them "gavincoins", but for the purpose of this argument that name is short and everyone here knows what it means) will not confirm on the other side of the fork where the 1MB limit is still followed.

So the market instantly realizes this difference between a Bitcoin and a GavinCoin.  So the value of a GavinCoin will drop relative to a Bitcoin.   Miners can't convert or spend these newly mined coins nearly anywhere, so all you have is buying from speculators.   Now those mining on the side which still recognizes the 1MB max limit are still mining blocks (albeit at a much slower rate because of the dramatic loss of hashing capacity) and some market for those newly mined coins exists -- again, thanks to speculators.    Things can flip quick.  Maybe a day goes by and all of a sudden a large amount of hashing capacity switches back to the 1MB max side due to the dropping exchange rate of GavinCoin, and it becomes quite possible (if not probable) that the 1MB limit will be with us for some time longer.

The exchanges and merchants that played both sides (i.e., required confirmations on both sides of the fork) lost nothing as either way they have confirmed transactions on what is eventually the sole winner.   If the hard fork fails then the losers are those who had E-Wallets (custodial accounts) and found their pre-fork bitcoins were spent and they only end up with tainted GavinCoins that can now never be spent (at least not anywhere near parity with a bitcoin).   Likely most every custodial service (e.g., exchanges, hosted/shared eWallets, etc.) that didn't require confirmations on both sides of the fork ends up bankrupt as a result of getting dumped on with GavinCoins while allowing withdrawals of untainted bitcoins.

I just don't see how a hard-fork succeeds.   There is risk of accepting GavinCoins.  There is no risk (excluding exchange rate risk) of putting your own pre-fork coins into storage for a (long) while and not letting them become tainted GavinCoins (as they can still be spent a year, two or ten later).

[Edited: A couple small readability changes.]