New Spaceship Speed in Conway’s Game of Life

In this article, I assume that you have basic familiarity with Conway’s Game of Life. If this is not the case, you can try reading an explanatory article but you will still struggle to understand much of the following content.

The day before yesterday ConwayLife.com forums saw a new member named zdr. When we the lifenthusiasts meet a newcomer, we expect to see things like “brand new” 30-cell 700-gen methuselah and then have to explain why it is not notable. However, what zdr showed us made our jaws drop.

It was a 28-cell c/10 orthogonal spaceship:

An animated image of the spaceship

To explain why this is such a groundbreaking discovery, I should first tell you that Life spaceships can be loosely divided into two categories. Engineered ships are the ones that consist of various small components. They often have adjustable speed. However, the population of tens of thousands to millions of cells causes these spaceships to have no practical value. There is much more incentive in hunting for elementary spaceships, which can be used for complex constructions. They are found using programs such as gfind or WLS. The algorithms behind these programs are beyond the scope of my article, but the important thing is that the search time goes up exponentially as the period of the ship grows. It is most interesting to find spaceships of new speeds, and the number of speeds that low-period ships can have is unfortunately limited:

  Orthogonal Diagonal
c/2 Yes Impossible
c/3 Yes Impossible
c/4 Yes Yes
c/5 Yes Yes
2c/5 Yes Impossible
c/6 Yes Yes
c/7 Yes Yes
2c/7 Yes Impossible
3c/7 No Impossible
c/8 No No
3c/8 No Impossible

This table does not include oblique speeds, which causes little inconvenience because no elementary oblique ships are known.

The table above shows that ships exist for most of possible speeds, and it seems obvious that the speeds for which there are no ships have been searched by numerous people with good knowledge of search programs, powerful computers and lots of patience. As for higher periods, even the smallest searches would take years on modern computers. It appears that low-hanging fruit have been harvested clean during the 46 years of Life research… or, more precisely, it appeared so before zdr’s post.

The idea we all missed is that if the ship is really microscopic, it can be found in reasonable time despite its high period. After zdr boldly went where no man has gone before, Josh Ball set up the corresponding search in gfind and refound the spaceship in a little over an hour. zdr said that their program found it in a matter of 19 seconds.

To be frank, similar event did happen before when the aforementioned Josh Ball pulled loafer out of a hat. However, zdr’s spaceship (which is now called Copperhead, as proposed by muzik) is much more awesome for a number of reasons:

  • Loafer is not so mind-bogglingly high-period.
  • Copperhead was much easier to find, so it is more surprising that nobody found it before.
  • Copperhead’s tail is relatively strong and can interact with other objects without breaking down.

The discovery of a new spaceship speed immediately opened a few new areas of research, which are being explored now.

Synthesis

Aidan F. Pierce came up with a Copperhead synthesis only 10 hours after the completion of the spaceship. The synthesis was inefficient, and a few people discovered better ones. The current best synthesis, made by Chris Cain, requires only 22 gliders. Its repeat time is 375 ticks, which means that a gun can start constructing the second spaceship 375 ticks after the first one. There is a 23-glider synthesis with a better repeat time of 373 ticks.

Incremental 22-glider synthesis of the copperhead

The synthesis can be substantially improved if we find this spaceship crawling out of a random soup. Adam P. Goucher has written a wonderful program called apgsearch, which is perfectly suited for this task. While the current version may be too slow to find a soup in reasonable time, highly anticipated version 3.0 can probably do the trick. Once it is found, it will appear here.

Guns

Once the synthesis was complete, building a gun was nothing but corollary-sniping. The first copperhead gun was created by myself, and a video of it is availible here. It was put together in a hurry and is therefore extremely inefficient. In particular, skilled gun builders can spot a silly mistake in the Northeast.

gmc_nxtman then made another gun with an almost optimal period of 376 ticks.

Eaters

simeks found two eaters for this ship, the better of which is shown below:

A copperhead eater

It is now time to search for a good copperhead-to-something-useful converter. The only existing one is clumsy and slow.

Sawtooths

Sawtooths often work by sending a flotilla of fast ships towards a slower ship. The more is the difference in speed, the less is the expansion factor of a sawtooth. Since expansion factor is proportional to how boring the sawtooth is, increasing the speed difference is a good thing. Dean Hickerson collided c/2 standard spaceships with c/10 copperhead to get a sawtooth with expansion factor 6:

Hickerson's sawtooth

He then made another sawtooth with expansion factor 10/3.

Puffers and rakes

Suppose a c/10 flotilla is hit by a glider. The glider turns into loads of mess, but all copperheads somehow survive and move on. The mess releases a glider, which flies into strategically placed second flotilla that is identical to the first one. Gliders continue to bounce back and forth between flotillas leaving mess behind them, and a c/10 puffer is complete! Unfortunately, this cool technique doesn’t work out easily in our case. There are no interesting interactions between a glider and a single copperhead, and it is unclear how one can place two or more copperheads so close to each other that a glider interacts with all of them. Assuming we figure it out, we can try to make a rake by perturbing the mess with copperheads so that it evolves into gliders, but that seems even less likely.

However, all this hand-waving can be turned it real puffers if we find…

Tagalongs

Tagalongs are small things that are attached to the back of a spaceship and move with it. Here is an example tagalong, called the Schick engine:

Schick engine pulled by two LWSSs

Finding a tagalong for the copperhead (or two copperheads) will be really nice. We can also try searching for pushalongs, but they are generally rarer.

Other patterns

There are a few other areas of Life exploration where the copperhead can be useful. For example, universal constructors often need to create an elbow still life very far away. It can be done by producing a copperhead, waiting for some time, and then shooting the copperhead down with a LWSS. At the moment I do not see why the copperhead can be better than the loafer in this aspect, but who knows?

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30 Responses to New Spaceship Speed in Conway’s Game of Life

  1. drc says:

    hey, that’s pretty good. Though, oblique spaceships have been observed in life, like the parallel hbk.

    Liked by 1 person

  2. A for awesome says:

    I like it a lot. One issue, though: it could be clearer that the optimized synthesis was not based on mine at all.

    Like

  3. Anonymous says:

    Obilque speeds do have known spaceships since 2010.

    Like

  4. mniemiec says:

    Back in the 1970s, Conway showed that a universal constructor was possible, implying that spaceships exist of all rational directions Engineered spaceships like Gemini, Caterpillar, and HBK demonstrate this. However, no ‘natural’ oblique spaceships have been found yet, although there have been some near misses, suggesting that small ones do likely exist.

    Like

  5. Alexey Nigin says:

    Thank you for pointing out the mistakes. I have edited the article accordingly.

    The Sawtooths section literally became obsolete overnight, so I amended it, too.

    Like

  6. Wright says:

    I’m confused… is the ‘silly mistake’ that you used two edge-shooters instead of a regular gun to start off the NE salvo?

    Like

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  11. Jeremy Tan says:

    I actually felt cheated when this came out. What is zdr’s real name, what is the background behind this user and what was the “19-second algorithm” used? This last point is particularly important, because if the algorithm is not posted then it’s as good as non-existent. (Case in point: a certain Hunt Chang has claimed to have solved the intersecting Bézier curve problem at https://sites.google.com/site/curvesintersection but he refuses to release the methods. I had to write one myself for my Kinross library.)

    Yes, I am Freywa on the ConwayLife forums. I used to be active there, but then I moved on to more continuous things: My Little Pony, Bézier curves, random number generation and now crosswords.

    Like

    • Alexey Nigin says:

      First of all, zdr has released the program that was used to find the copperhead. See this thread for details.

      Second of all, I can remember that you had a conversation with thunk about the use of real names on the Internet. Please don’t start it all over again here.

      All in all, I think one should not criticize a person who gave the CA community such an outstanding pattern.

      Like

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  14. Paul Wilson says:

    For some 46 years, even before home computers even, I have been doing albums of Life patterns. Some 32 books (incl. a Master Index). Hopefully now I have finished and can just read it all. Getting pages numbered neatly and correcting all transcription errors fixed ,plus the occasional new pattern (oscillators,spaceships and a few puffers, lately.). A lot of long-term ancestors and diehards before 2006…

    Like

  15. 6411muriel+1164 says:

    Eaters to destroy a copperhead (or some other painstakingly constructed spaceship, that doesn’t occur naturally)… How wasteful ! Better by far to use a Schick Flying Machine, a Cordership or a Coe-ship and create a puffer !

    Like

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  18. Paul Wilson says:

    I’ve seen on some websites C/2 spaceships that look like palm trees. Resolution is not good enough to copy pattern to paper and then to a program that runs the game. Some of these websites are archived and not being updated. (Like pentadecathlon.com )

    Like

    • Alexey Nigin says:

      I am not sure what you are talking about – with enough imagination, lots of patterns can look like palm trees. I encourage you to post the low-resolution images you mentioned, and then I will probably be able to find copypastable files of those patterns.

      Like

      • Paul Wilson says:

        You will find one at the very first image on the very first page of GoogleImages’ department of “Game of Life Line Puffers”. The shape COULD be painstakingly copied by hand onto graph paper, but it is large and with me,the larger a pattern,the more mistakes I might make. The pattern is C/3 P36 and outputs beehives. Pretty puny output for such a messy palmtree pattern. It even includes a C/3 P3 slipper spaceship,the simplest of these alien spaceships (don’t occur naturally *) at the tail.

        * I THOUGHT I had a possible appearing of this slipper spaceship in the output of a collision, but maybe I just dreamed of it.

        Like

        • Alexey Nigin says:

          Hello. As you can see, I respond to comments on my blog so quickly that I might qualify for the Quickest Responder Ever award administered by the Internet Sarcasm Department.

          Anyway, back to the topic. Here is the file of the pattern you were talking about:

          # P36 c/3 Beehive and Boat puffers
          #C by D.Bell
          x=262, y=231
          48b3o22b3o$46b2o3bo20bo3b2o109bo24bo$45bo3bo2bo18bo2bo3bo106b4o22b4o$45bo6b
          2obo12bob2o6bo105b2o3bo20bo3b2o$48b3ob2ob2o10b2ob2ob3o107bo5b3o16b3o5bo$45b
          3o3bob2o2bo8bo2b2obo3b3o107bobo4b2o10b2o4bobo$44b2o32b2o103bob4o4b2o10b2o4b
          4obo$56bo10bo114b8ob4o10b4ob8o$48b3o4b2o10b2o4b3o106b2o32b2o$39bo9b2o3b3o10b
          3o3b2o9bo102bo5b2o10b2o5bo$38bobob2o5b2o2b2o14b2o2b2o5b2obobo100bobo3bo2b
          o8bo2bo3bobo$37bo4b2o8b3o14b3o8b2o4bo90bo11bobo2bo10bo2bobo11bo$37bo3b3o6b
          o2bo16bo2bo6b3o3bo89b2ob3o5b4o18b4o5b3ob2o$52bo18bo103b2o5bo4b4o18b4o4bo5b
          2o$37b2ob2o13b2o10b2o13b2ob2o92bobo7bo2bo14bo2bo7bobo$36bobobo14b2ob2ob2ob
          2ob2o14bobobo87b2obo42bob2o$36bo3bob2o12bob2ob2ob2obo12b2obo3bo87b2ob2o13b
          3o8b3o13b2ob2o$37bo2bob2ob3o9b2o6b2o9b3ob2obo2bo87bobobobo15b2ob2ob2o15bob
          obobo$41bobo4b2o10bo2bo10b2o4bobo91bo3bob3obo8b2o4b2o4b2o8bob3obo3bo$46bob
          o8bo8bo8bobo100bo5b3o8b2ob4ob2o8b3o5bo$23bo9bob3o6bo3b3o7bo2b2o2bo7b3o3bo6b
          3obo9bo78bob3o2b2o7b2o6b2o7b2o2b3obo$22b2o9b2ob2o7bob2obob3o5bo2bo5b3obob
          2obo7b2ob2o9b2o72bo13bo10b2o10bo13bo$13b2o3b2obobo9bo4bo2b4o8b2o4b2o2b2o4b
          2o8b4o2bo4bo9bobob2o3b2o49b2o9bobobo7b2o3b2obo7b2o7bob2o3b2o7bobobo9b2o$10b
          2ob2o2bo3bo2bob2ob2obo2bo8bob3obo2b2o4b2o2b2o4b2o2bob3obo8bo2bob2ob2obo2b
          o3bo2b2ob2o47b2o7b2o4bo3bo2bob2ob3obo5bo2bo5bob3ob2obo2bo3bo4b2o7b2o$7b4o2b
          o2bo4b4o2bobo2b4o2bo5bob2ob2ob4ob2o6b2ob4ob2ob2obo5bo2b4o2bobo2b4o4bo2bo2b
          4o33b3o3bob2ob2o7b2ob3o4b3o6bo3bo5b2o5bo3bo6b3o4b3ob2o7b2ob2obo3b3o$6bo4b
          o4b2o5b2o2bobo2b2o4b2ob5o4bobo4bobob2obobo4bobo4b5ob2o4b2o2bobo2b2o5b2o4b
          o4bo28bob2o3bob2o2bo2b4ob4o2bo6bobobobob2o8bo2bo8b2obobobobo6bo2b4ob4o2bo2b
          2obo3b2obo$7b2o7b2o23bobo5b2obo4bobo4bobo4bob2o5bobo23b2o7b2o28b3o3b4o4bo5b
          obo5bo2b2o2bo7bobo2b3o3b2o3b3o2bobo7bo2b2o2bo5bobo5bo4b4o3b3o$53bob2obobo2b
          obob2obo73bo3bo4bo8bo7bo5b4o3b2obo3bobo2bob2o2b2obo2bobo3bob2o3b4o5bo7bo8b
          o4bo3bo$53b3o12b3o74bo8b2o22b2obo5b2ob2o3bobob2obobo3b2ob2o5bob2o22b2o8bo$22b
          o9bo20b2o2bo3b2o3bo2b2o20bo9bo88b2ob5o4b5ob2o$21b4o5b3ob3o18b2o2b6o2b2o18b
          3ob3o5b4o87bo4bo3b2o3bo4bo$8bo7b4o4b2o3b2o6bo10b2ob2o3bobo6bobo3b2ob2o10b
          o6b2o3b2o4b4o7bo43b2o9b2obo17bo16bo17bob2o9b2o$b2ob3ob3o5bobobobo3bobo3bob
          obo3b3o4bobob2o18b2obobo4b3o3bobobo3bobo3bobobobo5b3ob3ob2o32b2o2b2ob2o3b
          2ob2ob2o15b5ob3o2b3ob5o15b2ob2ob2o3b2ob2o2b2o$b2o4bo2b2ob2o2bo2b5obobo3bob
          2o3b2ob4obobobo5b2o6b2o5bobobob4ob2o3b2obo3bobob5o2bo2b2ob2o2bo4b2o20bob2o7b
          ob3o3b2o3b2obobo2bo3bo6b2ob2o2b3ob6ob3o2b2ob2o6bo3bo2bobob2o3b2o3b3obo7b2ob
          o$o2bob2o3bo2b2o5b2ob2obobobobobo4b2o3b2o32b2o3b2o4bobobobobob2ob2o5b2o2b
          o3b2obo2bo15b3ob2obob2o3b2o3bobo3bobo2bo2bobo2b2obobob2obo8bo6bo8bob2obob
          ob2o2bobo2bo2bobo3bobo3b2o3b2obob2ob3o$12bo6bobo3bo3bobo2bob2obo3bob3o11b
          6o11b3obo3bob2obo2bobo3bo3bobo6bo26bo6b2o2b5obo6bobobo3bobo7bo2bo5b2o18b2o5b
          o2bo7bobo3bobobo6bob5o2b2o6bo$19bo3bo10bob2o8bobo26bobo8b2obo10bo3bo34b2o3b
          o3bo3bo4bo6bobobobobobo4b2o4b2o9b8o9b2o4b2o4bobobobobobo6bo4bo3bo3bo3b2o$21b
          o8bo4bob2o4b2o3bo4b2o14b2o4bo3b2o4b2obo4bo8bo48bo5bobobobo3b3o2bob2ob2o6b
          o12b4o12bo6b2ob2obo2b3o3bobobobo5bo$30bo8b3obo6b2ob2o14b2ob2o6bob3o8bo66b
          o7bo3bo8bo2bobo26bobo2bo8bo3bo7bo$39bobo2b2obo2b2o3bo2bo6bo2bo3b2o2bob2o2b
          obo88bob4ob3o2bobo2b3o6b2o6b3o2bobo2b3ob4obo$45bob2o2bo4b2o2bo2bo2b2o4bo2b
          2obo98bobobobo3b3o3bo5b2o5bo3b3o3bobobobo$40b3obo12bo8bo12bob3o93bob2ob2ob
          3o3b2o14b2o3b3ob2ob2obo$57b2o6b2o113b2obob2ob2o20b2ob2obob2o$59bob2obo114b
          o40bo$61b2o116bo16bobo2bobo16bo$61b2o133bobo2bobo$195bo8bo$39b2o16bo8bo129b
          8o$36b2ob2o15b3o6b3o128b8o$33b4o2b2o4b2o8bo3bo4bo3bo125b3o6b3o$32bo4bo4b2ob
          2o8b2ob2o4b2ob2o124bo3bo4bo3bo$33b2o7b2o3bo144bo5bo2bo5bo$42b2o148bo5bo2b
          o5bo15b3ob2o$176b3o13b3ob3o2b3ob3o13b2o4b2o$172bob2o3bo40bo3bobo2bo$171b3o3b
          3o2b3o30b3o2b3o$170bo3bo4b3obobo28bobob3o$171bo7bo4bo30bo4bo$46bo104b2o27b
          2o36b2o$45bobo97bo5b2ob2o5b2o26b2o18b2o20bo11b4o$45bobo96b3o3bo3b2o4bob2o24b
          o2bo16bo2bo12b2o4b3obo8bobob2obo$46bo96bo3bo6b2o3bo2b2o2b3o2b3obo13b2o18b
          2o13b2ob2obob2obo3b3o2bo4b3o$144bobobobobo5bo2bob3o3bob3ob2o46bo3bo4bo3bob
          ob2o6bo3bo$150bobo3b4ob2o3b5o2bo3bo43bo4b3o4bob3o2b2o9b2o$150bobo3b2o3b2o3b
          2o6bo2bo42b2o3b2ob2o7bo2bobo8b2o$152bo13bo3b3o2bob2o39b2o2b2obobo3bo8bobob
          2o5bob2o$162bo3bo7b2obobo38b3ob2obo2bo4b3o2bobobob2o4b2obob2o$172bo2bobob
          2o38b2ob2obobo2bobo2b4obo4bo11bo$174bo2bo51bo6b2o5bo3b2ob2ob2obo$173b3obob
          2o7b3o18b3o10bo13bo3b4o3bo$46bo128bo47b2o12bo10b2o$45bobo127bo6b2o5bo20bo4b
          3o20bo$45bobo137bobobo20bobob2obo$46bo135bo7bo18bo$183b3o4bo18bo4b2o$184b
          obo4bo16bo4bobo$187bo3bo16bo3bo$188b4o16b4o$196bo$196bo$196bo2$46bo145b3o3b
          3o$45bobo$45bobo136b2o10bo$46bo137b3o9bo$186b2o2b2o4bo$186bobo2bo$191bo$186b
          o2bo$180b2o$181b2obo2bob3o$181bo3bobo3bo$181b3obobobo$181bo3bobo$181b3o3b
          2o$183b4o$185bo2$192b3o$191b2ob2o$190bobob2o$190bo3bo$191bo2bo$191b3o2$167b
          2o$164b2ob2o$161b4o2b2o4b2o$160bo4bo4b2ob2o4bo12b2o$161b2o7b2o3bo2bo6b2o5b
          2o$170b2o6bobo3bobo$178bobo$183bob2o$182b2obo$182b5o$181bo3b2o$181bo2bo6b
          o4bo$182b3o5bobob4o$190bo2bo4bo$191b3ob5o$194b2o2bo$196b2o2$216bo$215b4o$214b
          o3b2o$212b3o5bo$185b2o8bo13b2o4bobo$181b2o2b2o8b3o11b2o4b4obo$181b2o11bo3b
          o10b4ob8o$191b3o4bo21b2o$191b3obob2o10b2o5bo$191b2o3bo11bo2bo3bobo$209bo2b
          obo$213b4o$213b4o$205b3o3bo2bo$205bob2o$206bob2o$185b2o19bobo$181b2o2b2o14b
          o4b2o$181b2o18bobo$190b2o9b2o$190b2o8$185b2o$181b2o2b2o$181b2o$190b2o$190b
          2o6$192bo$192bobo9bo$185b2o5b2o9b4o$181b2o2b2o11bo3bo3b2ob2o$181b2o14b4o5b
          o2b2o$190b2o4bo3bo7bo2bo$190b2o5bobo2bo$201bo2$184bo$184bo$183bobo$181b2o$181b
          o$183bo3bo$182b2o3bo$180b3o4bo7$166b3o$162bob2o3bo$161b3o3b3o2b3o10bo$160b
          o3bo4b3o3bo9bo$161bo7bo3b2o10bo$170b2o9b3o$183bo$179bo3bo$178bo2b2o2$178b
          2o9$181b2o$181bobo$182bo10$181b2o$181bobo$182bo10$181b2o$181bobo$182bo!

          It is in RLE format, which can be read by nearly every CA program in existence. If the program you are using doesn’t properly decode it, you should seriously consider installing Golly.

          Regarding your other point, puffers are intentionally engineered to produce as little output as possible, because then they are way more useful in the construction of larger patterns. And no, there is no way you could see any c/3 ship coming out of a random soup unless you made a huge distributed search, and even then it would have been extremely unlikely.

          Like

      • Paul Wilson says:

        At the mo, I have created a 32nd -and last- volume scrapbook. It has 2 greyships no less, and the Boojum, and some other exotic stuff. I did include the Palm Tree Puffer in Vol.26 (body is P2, but it takes 3 periods to output a block. The blocks form a long “stem”.

        The greyships were troublesome to copy without errors. One I call the Angler Fish, as it has LW classic spaceships,imbedded in the trailing end. (like the degenerate males in deep-sea species)and moving as if they were alone (resorbed and then reappear).

        I find from dates appended to most patterns that I started with Life in 1970, began these archive albums in 1978 continued to 2000,then paused until 2006, which a cellar flood destroyed most of my work -but not the Master Index, and some early albums stored on a higher shelf. Repair work segued into new stuff entirely (oscillators,spaceships, puffers, guns, factories, stretchers,…), As of June, 2017 I expect to have finally finished a grand archive.

        Like

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  20. Paul Wilson says:

    That LIFE websites seem to be archived and no longer updated, does not encourage me to continue.

    Like

    • Alexey Nigin says:

      Neither Pentadecathlon nor my blog should serve as an indication of the overall health of the CA community. If you go to ConwayLife forums, you will find a lot of people interested to hear about your work. If you have finished assembling your collection, you should definitely post it there.

      Like

      • Paul Wilson says:

        I surfed the forums, but gave up. Nobody posts pics of their work, just golly gibberish. A simple plotting of a pic is all I asked.

        As I said, I no longer add patterns, but carefully check page-numbering ,indexing, and master index (vo1.0) numbering. I found 1 duplication, and a few vol.ref.errors. I also replaced pages that were blotched (an older printer darkened faint blotches into big messes), and fixed some transcription bungles, cross-reference errors, as well as age-danaged oages, (Vol.1 has NO original pgs left, and only 2 orig,pages in vol.2.).

        I did find 18 gliders (glide-reflection halfway through period) and one was P6 (Seal. A forum writer mentioned that.Thanks). One I created (extendible). None of these were distinguished as gliders from other diagonal spaceships. I(ich) made the distinction. I !
        Just as I found 40 or so Conservative Reactjons (my term. Two or more patterns that collide to reappear rearranged, but with no other stuff in arena,xcept a bit iof junk).

        All this is accomplished. I no longer research, After 47 years…

        Like

  21. Paul Wilson says:

    The Schick Engine I see as a failed puffer,as the junk exhaust lasts long enough that it lends itself to being vitalized into lasting exhaust. So my albums list it under “puffer trains” not “spaceships”.

    SPARK – A 1- or 2- bit fragment that dies in 1 gen.
    JUNK – A 3- bit or larger fragment that takes 2 or more gen. to die.
    DIEHARD – A fragment that takes long enough to die that its longeivity/size ratio is 7 or higher. I have charted almost 200 of these, a few going more than 200 gen.

    Like

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