📜 [專欄新文章] Uniswap v3 Features Explained in Depth
✍️ 田少谷 Shao
📥 歡迎投稿: https://medium.com/taipei-ethereum-meetup #徵技術分享文 #使用心得 #教學文 #medium
Once again the game-changing DEX 🦄 👑
Image source: https://uniswap.org/blog/uniswap-v3/
Outline
0. Intro1. Uniswap & AMM recap2. Ticks 3. Concentrated liquidity4. Range orders: reversible limit orders5. Impacts of v36. Conclusion
0. Intro
The announcement of Uniswap v3 is no doubt one of the most exciting news in the DeFi place recently 🔥🔥🔥
While most have talked about the impact v3 can potentially bring on the market, seldom explain the delicate implementation techniques to realize all those amazing features, such as concentrated liquidity, limit-order-like range orders, etc.
Since I’ve covered Uniswap v1 & v2 (if you happen to know Mandarin, here are v1 & v2), there’s no reason for me to not cover v3 as well ✅
Thus, this article aims to guide readers through Uniswap v3, based on their official whitepaper and examples made on the announcement page. However, one needs not to be an engineer, as not many codes are involved, nor a math major, as the math involved is definitely taught in your high school, to fully understand the following content 😊😊😊
If you really make it through but still don’t get shxt, feedbacks are welcomed! 🙏
There should be another article focusing on the codebase, so stay tuned and let’s get started with some background noise!
1. Uniswap & AMM recap
Before diving in, we have to first recap the uniqueness of Uniswap and compare it to traditional order book exchanges.
Uniswap v1 & v2 are a kind of AMMs (automated market marker) that follow the constant product equation x * y = k, with x & y stand for the amount of two tokens X and Y in a pool and k as a constant.
Comparing to order book exchanges, AMMs, such as the previous versions of Uniswap, offer quite a distinct user experience:
AMMs have pricing functions that offer the price for the two tokens, which make their users always price takers, while users of order book exchanges can be both makers or takers.
Uniswap as well as most AMMs have infinite liquidity¹, while order book exchanges don’t. The liquidity of Uniswap v1 & v2 is provided throughout the price range [0,∞]².
Uniswap as well as most AMMs have price slippage³ and it’s due to the pricing function, while there isn’t always price slippage on order book exchanges as long as an order is fulfilled within one tick.
In an order book, each price (whether in green or red) is a tick. Image source: https://ftx.com/trade/BTC-PERP
¹ though the price gets worse over time; AMM of constant sum such as mStable does not have infinite liquidity
² the range is in fact [-∞,∞], while a price in most cases won’t be negative
³ AMM of constant sum does not have price slippage
2. Tick
The whole innovation of Uniswap v3 starts from ticks.
For those unfamiliar with what is a tick:
Source: https://www.investopedia.com/terms/t/tick.asp
By slicing the price range [0,∞] into numerous granular ticks, trading on v3 is highly similar to trading on order book exchanges, with only three differences:
The price range of each tick is predefined by the system instead of being proposed by users.
Trades that happen within a tick still follows the pricing function of the AMM, while the equation has to be updated once the price crosses the tick.
Orders can be executed with any price within the price range, instead of being fulfilled at the same one price on order book exchanges.
With the tick design, Uniswap v3 possesses most of the merits of both AMM and an order book exchange! 💯💯💯
So, how is the price range of a tick decided?
This question is actually somewhat related to the tick explanation above: the minimum tick size for stocks trading above 1$ is one cent.
The underlying meaning of a tick size traditionally being one cent is that one cent (1% of 1$) is the basis point of price changes between ticks, ex: 1.02 — 1.01 = 0.1.
Uniswap v3 employs a similar idea: compared to the previous/next price, the price change should always be 0.01% = 1 basis point.
However, notice the difference is that in the traditional basis point, the price change is defined with subtraction, while here in Uniswap it’s division.
This is how price ranges of ticks are decided⁴:
Image source: https://uniswap.org/whitepaper-v3.pdf
With the above equation, the tick/price range can be recorded in the index form [i, i+1], instead of some crazy numbers such as 1.0001¹⁰⁰ = 1.0100496621.
As each price is the multiplication of 1.0001 of the previous price, the price change is always 1.0001 — 1 = 0.0001 = 0.01%.
For example, when i=1, p(1) = 1.0001; when i=2, p(2) = 1.00020001.
p(2) / p(1) = 1.00020001 / 1.0001 = 1.0001
See the connection between the traditional basis point 1 cent (=1% of 1$) and Uniswap v3’s basis point 0.01%?
Image source: https://tenor.com/view/coin-master-cool-gif-19748052
But sir, are prices really granular enough? There are many shitcoins with prices less than 0.000001$. Will such prices be covered as well?
Price range: max & min
To know if an extremely small price is covered or not, we have to figure out the max & min price range of v3 by looking into the spec: there is a int24 tick state variable in UniswapV3Pool.sol.
Image source: https://uniswap.org/whitepaper-v3.pdf
The reason for a signed integer int instead of an uint is that negative power represents prices less than 1 but greater than 0.
24 bits can cover the range between 1.0001 ^ (2²³ — 1) and 1.0001 ^ -(2)²³. Even Google cannot calculate such numbers, so allow me to offer smaller values to have a rough idea of the whole price range:
1.0001 ^ (2¹⁸) = 242,214,459,604.341
1.0001 ^ -(2¹⁷) = 0.000002031888943
I think it’s safe to say that with a int24 the range can cover > 99.99% of the prices of all assets in the universe 👌
⁴ For implementation concern, however, a square root is added to both sides of the equation.
How about finding out which tick does a price belong to?
Tick index from price
The answer to this question is rather easy, as we know that p(i) = 1.0001^i, simply takes a log with base 1.0001 on both sides of the equation⁴:
Image source: https://www.codecogs.com/latex/eqneditor.php
Let’s try this out, say we wanna find out the tick index of 1000000.
Image source: https://ncalculators.com/number-conversion/log-logarithm-calculator.htm
Now, 1.0001¹³⁸¹⁶² = 999,998.678087146. Voila!
⁵ This formula is also slightly modified to fit the real implementation usage.
3. Concentrated liquidity
Now that we know how ticks and price ranges are decided, let’s talk about how orders are executed in a tick, what is concentrated liquidity and how it enables v3 to compete with stablecoin-specialized DEXs (decentralized exchange), such as Curve, by improving the capital efficiency.
Concentrated liquidity means LPs (liquidity providers) can provide liquidity to any price range/tick at their wish, which causes the liquidity to be imbalanced in ticks.
As each tick has a different liquidity depth, the corresponding pricing function x * y = k also won’t be the same!
Each tick has its own liquidity depth. Image source: https://uniswap.org/blog/uniswap-v3/
Mmm… examples are always helpful for abstract descriptions 😂
Say the original pricing function is 100(x) * 1000(y) = 100000(k), with the price of X token 1000 / 100 = 10 and we’re now in the price range [9.08, 11.08].
If the liquidity of the price range [11.08, 13.08] is the same as [9.08, 11.08], we don’t have to modify the pricing function if the price goes from 10 to 11.08, which is the boundary between two ticks.
The price of X is 1052.63 / 95 = 11.08 when the equation is 1052.63 * 95 = 100000.
However, if the liquidity of the price range [11.08, 13.08] is two times that of the current range [9.08, 11.08], balances of x and y should be doubled, which makes the equation become 2105.26 * 220 = 400000, which is (1052.63 * 2) * (110 * 2) = (100000 * 2 * 2).
We can observe the following two points from the above example:
Trades always follow the pricing function x * y = k, while once the price crosses the current price range/tick, the liquidity/equation has to be updated.
√(x * y) = √k = L is how we represent the liquidity, as I say the liquidity of x * y = 400000 is two times the liquidity of x * y = 100000, as √(400000 / 100000) = 2.
What’s more, compared to liquidity on v1 & v2 is always spread across [0,∞], liquidity on v3 can be concentrated within certain price ranges and thus results in higher capital efficiency from traders’ swapping fees!
Let’s say if I provide liquidity in the range [1200, 2800], the capital efficiency will then be 4.24x higher than v2 with the range [0,∞] 😮😮😮 There’s a capital efficiency comparison calculator, make sure to try it out!
Image source: https://uniswap.org/blog/uniswap-v3/
It’s worth noticing that the concept of concentrated liquidity was proposed and already implemented by Kyper, prior to Uniswap, which is called Automated Price Reserve in their case.⁵
⁶ Thanks to Yenwen Feng for the information.
4. Range orders: reversible limit orders
As explained in the above section, LPs of v3 can provide liquidity to any price range/tick at their wish. Depending on the current price and the targeted price range, there are three scenarios:
current price < the targeted price range
current price > the targeted price range
current price belongs to the targeted price range
The first two scenarios are called range orders. They have unique characteristics and are essentially fee-earning reversible limit orders, which will be explained later.
The last case is the exact same liquidity providing mechanism as the previous versions: LPs provide liquidity in both tokens of the same value (= amount * price).
There’s also an identical product to the case: grid trading, a very powerful investment tool for a time of consolidation. Dunno what’s grid trading? Check out Binance’s explanation on this, as this topic won’t be covered!
In fact, LPs of Uniswap v1 & v2 are grid trading with a range of [0,∞] and the entry price as the baseline.
Range orders
To understand range orders, we’d have to first revisit how price is discovered on Uniswap with the equation x * y = k, for x & y stand for the amount of two tokens X and Y and k as a constant.
The price of X compared to Y is y / x, which means how many Y one can get for 1 unit of X, and vice versa the price of Y compared to X is x / y.
For the price of X to go up, y has to increase and x decrease.
With this pricing mechanism in mind, it’s example time!
Say an LP plans to place liquidity in the price range [15.625, 17.313], higher than the current price of X 10, when 100(x) * 1000(y) = 100000(k).
The price of X is 1250 / 80 = 15.625 when the equation is 80 * 1250 = 100000.
The price of X is 1315.789 / 76 = 17.313 when the equation is 76 * 1315.789 = 100000.
If now the price of X reaches 15.625, the only way for the price of X to go even higher is to further increase y and decrease x, which means exchanging a certain amount of X for Y.
Thus, to provide liquidity in the range [15.625, 17.313], an LP needs only to prepare 80 — 76 = 4 of X. If the price exceeds 17.313, all 4 X of the LP is swapped into 1315.789 — 1250 = 65.798 Y, and then the LP has nothing more to do with the pool, as his/her liquidity is drained.
What if the price stays in the range? It’s exactly what LPs would love to see, as they can earn swapping fees for all transactions in the range! Also, the balance of X will swing between [76, 80] and the balance of Y between [1250, 1315.789].
This might not be obvious, but the example above shows an interesting insight: if the liquidity of one token is provided, only when the token becomes more valuable will it be exchanged for the less valuable one.
…wut? 🤔
Remember that if 4 X is provided within [15.625, 17.313], only when the price of X goes up from 15.625 to 17.313 is 4 X gradually swapped into Y, the less valuable one!
What if the price of X drops back immediately after reaching 17.313? As X becomes less valuable, others are going to exchange Y for X.
The below image illustrates the scenario of DAI/USDC pair with a price range of [1.001, 1.002] well: the pool is always composed entirely of one token on both sides of the tick, while in the middle 1.001499⁶ is of both tokens.
Image source: https://uniswap.org/blog/uniswap-v3/
Similarly, to provide liquidity in a price range < current price, an LP has to prepare a certain amount of Y for others to exchange Y for X within the range.
To wrap up such an interesting feature, we know that:
Only one token is required for range orders.
Only when the current price is within the range of the range order can LP earn trading fees. This is the main reason why most people believe LPs of v3 have to monitor the price more actively to maximize their income, which also means that LPs of v3 have become arbitrageurs 🤯
I will be discussing more the impacts of v3 in 5. Impacts of v3.
⁷ 1.001499988 = √(1.0001 * 1.0002) is the geometric mean of 1.0001 and 1.0002. The implication is that the geometric mean of two prices is the average execution price within the range of the two prices.
Reversible limit orders
As the example in the last section demonstrates, if there is 4 X in range [15.625, 17.313], the 4 X will be completely converted into 65.798 Y when the price goes over 17.313.
We all know that a price can stay in a wide range such as [10, 11] for quite some time, while it’s unlikely so in a narrow range such as [15.625, 15.626].
Thus, if an LP provides liquidity in [15.625, 15.626], we can expect that once the price of X goes over 15.625 and immediately also 15.626, and does not drop back, all X are then forever converted into Y.
The concept of having a targeted price and the order will be executed after the price is crossed is exactly the concept of limit orders! The only difference is that if the range of a range order is not narrow enough, it’s highly possible that the conversion of tokens will be reverted once the price falls back to the range.
As price ranges follow the equation p(i) = 1.0001 ^ i, the range can be quite narrow and a range order can thus effectively serve as a limit order:
When i = 27490, 1.0001²⁷⁴⁹⁰ = 15.6248.⁸
When i = 27491, 1.0001²⁷⁴⁹¹ = 15.6264.⁸
A range of 0.0016 is not THAT narrow but can certainly satisfy most limit order use cases!
⁸ As mentioned previously in note #4, there is a square root in the equation of the price and index, thus the numbers here are for explantion only.
5. Impacts of v3
Higher capital efficiency, LPs become arbitrageurs… as v3 has made tons of radical changes, I’d like to summarize my personal takes of the impacts of v3:
Higher capital efficiency makes one of the most frequently considered indices in DeFi: TVL, total value locked, becomes less meaningful, as 1$ on Uniswap v3 might have the same effect as 100$ or even 2000$ on v2.
The ease of spot exchanging between spot exchanges used to be a huge advantage of spot markets over derivative markets. As LPs will take up the role of arbitrageurs and arbitraging is more likely to happen on v3 itself other than between DEXs, this gap is narrowed … to what extent? No idea though.
LP strategies and the aggregation of NFT of Uniswap v3 liquidity token are becoming the blue ocean for new DeFi startups: see Visor and Lixir. In fact, this might be the turning point for both DeFi and NFT: the two main reasons of blockchain going mainstream now come to the alignment of interest: solving the $$ problem 😏😏😏
In the right venue, which means a place where transaction fees are low enough, such as Optimism, we might see Algo trading firms coming in to share the market of designing LP strategies on Uniswap v3, as I believe Algo trading is way stronger than on-chain strategies or DAO voting to add liquidity that sort of thing.
After reading this article by Parsec.finance: The Dex to Rule Them All, I cannot help but wonder: maybe there is going to be centralized crypto exchanges adopting v3’s approach. The reason is that since orders of LPs in the same tick are executed pro-rata, the endless front-running speeding-competition issue in the Algo trading world, to some degree, is… solved? 🤔
Anyway, personal opinions can be biased and seriously wrong 🙈 I’m merely throwing out a sprat to catch a whale. Having a different voice? Leave your comment down below!
6. Conclusion
That was kinda tough, isn’t it? Glad you make it through here 🥂🥂🥂
There are actually many more details and also a huge section of Oracle yet to be covered. However, since this article is more about features and targeting normal DeFi users, I’ll leave those to the next one; hope there is one 😅
If you have any doubt or find any mistake, please feel free to reach out to me and I’d try to reply AFAP!
Stay tuned and in the meantime let’s wait and see how Uniswap v3 is again pioneering the innovation of DeFi 🌟
Uniswap v3 Features Explained in Depth was originally published in Taipei Ethereum Meetup on Medium, where people are continuing the conversation by highlighting and responding to this story.
👏 歡迎轉載分享鼓掌
同時也有2部Youtube影片,追蹤數超過15萬的網紅jeonsann,也在其Youtube影片中提到,this was one year ago lol ♡ follow me for updates ♡ ↬ main channel: https://www.youtube.com/c/jeonsann ↬ idk what this channel is for anymore: https...
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【2019.07區塊鏈動態】
區塊鏈動起來,金融、貨運、製造業都來參一腳!
睜大眼睛看區塊鏈發生什麼事👀
(一)以瑞銀為首的14家金融機構合資成立公司,開發區塊鏈金融基礎架構
以瑞銀為首的14家大型金融機構,共同創立名為Fnality International的公司,合計投資Fnality 5,000萬英鎊,目的是管理「結算專用幣」(utility settlement coin;USC)的開發。
(二)全球六大航運商已有五家加入IBM區塊鏈貨櫃追蹤平台
於7月初,IBM 宣布,德國赫伯羅特 (Hapag-Lloyd)、日本新集運(Ocean Network Express)等兩家全球主要航運商加入其區塊鏈貨櫃追蹤平台TradeLens,至此,全球前六大航運商已經有五家使用 TradeLens,這代表著該平台所追蹤的貨櫃,已經超過全球運量一半。
(三)100家日本製造商利用區塊鏈來共享數據
據日經亞洲評論報導,明年春天,日本製造企業將開始使用區塊鏈共享數據,參與者包括三菱電子、日立、工具製造商DMG MORI以及製造機器人公司FANUC和Yaskawa Electric等100家企業,該項目得到日本經濟產業省(METI)的支持。
(四)Digital Asset 宣布將DAML智能合約語言與Hyperledger Fabric、R3 Corda整合
DAML智能合約語言是由 Digital Asset 所開發的,旨在縮減多方交易的業務流程。DAML 將簡化區塊鏈平台底層的複雜性,允許開發人員專心在業務流程的應用程式邏輯,同時為客戶提供在多種區塊鏈上部署相關的應用程式。在此之前DAML已經和區塊鏈平台 VMware Blockchain 和 Hyperledger Sawtooth 整合完成。
(五)「台灣區塊鏈大聯盟」於7月12日啟動
國發會從2月開始就已展開臺灣區塊鏈大聯盟的籌備工作,期間邀集國內區塊鏈相關各界代表,包括臺灣金融科技協會、奧丁丁集團、AMIS帳聯網等業者;金管會、經濟部、科技部、教育部等政府部門;台大、交大、文化大學等學者,以及倡議甚力的立法委員余宛如、許毓仁等,進行了2場座談會及2次籌備會。
與會代表一致同意有必要成立此一聯盟,做為業者、政府及學研機構的溝通平台,並整合各界資源,暢通資訊交流,推動法規調適、國際合作、場域應用、人才培育等工作,創造良好的區塊鏈生態圈。
http://sc.piee.pw/K29GK
#區塊鏈 #加密貨幣 #支付
ocean coin 在 葡萄牙自由行 Uma Volta em Portugal Facebook 的最讚貼文
【一點有趣的葡萄牙歷史 ~~《莫三比克的鈔票》】【A little bit of the history of Portugal from 《A banknote of Mozambique》】
大家可能覺得有點奇怪, 何解我會拿張鈔票出來呢? 跟葡萄牙有怎麼關係呢? 其實是這樣的: 我有一位好友是錢幣收藏家, 早陣子送我這張鈔票 (因為他知道我喜好跟葡萄牙有關的東西)。
You may wonder why I am showing you this banknote. What's it to do with Portugal? Indeed, a good friend of mine is a coin/paper money collector. He gave me this Mozambique banknote because he knows I am a "Luso-lover".
莫三比克 (舊稱: 莫三鼻給) 位處非洲東南部,臨印度洋,隔著莫三比克海峽 (Mozambique Channel) 與馬達加斯加 (Madagascar) 遙遙相望。莫三比克以葡萄牙語作為其官方語言, 皆因早在1498年, 葡萄牙的著名航海家達伽瑪 (Vasco da Gama) 就率領船隊到達莫三比克。 到1505年時, 莫三比克更成為了葡萄牙的殖民地, 歷時超過四百年, 直到1975年6月25日從葡萄牙獨立, 成為「莫三比克人民共和國」。
Mozambique is a former Portuguese colony. Situated in southeast Africa, it is bordered by the Indian Ocean to the east and separated from Madagascar by the Mozambique Channel. Mozambique was explored by the great Portuguese explorer Vasco da Gama in 1498 and was then colonised by Portugal in 1505. After over four centuries of Portuguese colonial rule, Mozambique became independent from Portugal on June 25, 1975, becoming the People's Republic of Mozambique shortly thereafter.
我擁有的這張莫三比克鈔票 (前和背面), 印刷於1970年 (當時莫三比克還是葡萄牙的殖民地), 背面印著發鈔銀行是大西洋銀行 (葡萄牙國家海外銀行) (Banco Nacional Ultramarino), 幣值 50 escudos (埃斯庫多, 是葡萄牙未使用歐羅之前所使用的貨幣單位)。
但有點特別的地方是: 鈔票正面卻又印著 "Banco de Moçambique" (莫三比克銀行) ? 為怎麼會這樣的呢? 莫三比克在1975年獨立後,新政府發現在他們的金庫裡頭有大量非流通的葡萄牙殖民時代的莫三比克鈔票。由於新政府需盡快有新的貨幣,於是就將 "Banco de Moçambique" 這名稱 "叠印“在這些本為大西洋銀行印刷發行的鈔票之上, 成為有趣的過渡時期鈔票。
This Mozambique note that I have (front and back) was printed in Year 1970 (at the time Mozambique was still a colony of Portugal). In the back of it, it shows "Banco Nacional Ultramarino" is the issuing bank. The value of the note is 50 escudos (escudos was the currency that Portugal used prior to the euro).
One interesting thing is: "Banco de Moçambique" (Bank of Mozambique) is printed on the front of the note as well? Why? After becoming independent in 1975, the new Mozambique government found a large amount of uncirculated Portuguese Mozambique notes in their vaults. Needing new currency quickly, they overprinted “Banco de Mocambique” on the old notes that had originally produced for the Banco Nacional Ultramarino. This overprinted note that I have, for example, was used by Mozambique between 1975 and 1980, when a new series of notes was finally released. It is an interesting transitional currency.
附加: 根據歷史推算, 當年鄭和下西洋 (1405-1433年), 他的船隊的終極點很大可能在今天的莫三比克! :-)
P.S.: According to historical projections, Zheng He - the great Admiral from the Ming Dynasty of China - could have reached as far as today's Mozambique in his fleet in the huge Zheng-He Expeditions (1405-1433)! :-)
ocean coin 在 jeonsann Youtube 的最讚貼文
this was one year ago lol
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ocean coin 在 translation Youtube 的最讚貼文
#セガ第1AM研究開発部が開発、セガが稼働させたAC用レースゲームのシリーズ第3弾且つセガ最後の非ポリゴンレーシング作。
主な特徴としては、最大8人までの通信対戦が可能、選択車種が8種類に増加、ルート分岐が復活等が挙げられる。
シリーズの中で唯一AM1研で制作され、初代アウトランのスタッフも何名か携わっているが鈴木裕氏は無関与。
BGMはHiro師匠、バーチャファイターシリーズ等の中村氏、デイトナUSAシリーズやセガラリーシリーズ等の光吉氏といったSEGA所属の3名による共同作・編曲。初代アウトランのBGMもアレンジされ使用されている。
作・編曲:川口博史氏、中村隆之氏、光吉猛修氏
Year: 1993.05.15(J),1993.06.12(U),1993.07.11(EU)
Manufacturer: SEGA / sega AM1 R&D
System: System32 multi
Hardware: Z80,YM3438,SEGA Multi pcm
Composer: hiroshi kawaguchi,takayuki nakamura, takenobu mitsuyoshi
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00:00:00 01.Coin (クレジット)
00:00:01 02.Engine Start (スタート / エンジン音)
00:00:04 03.Mega Driver (メガドライバー)
00:01:17 04.Magical Sound Shower (1993) (Speed BusterのデフォルトBGM)
00:05:56 05.Passing Breeze (1993)
00:10:29 06.Splash Wave (1993) (Mad PowerのデフォルトBGM)
00:14:52 07.Picture the Rivers (Smooth OperatorのデフォルトBGM)
00:18:08 08.Blow Your Cool (Bad BoyのデフォルトBGM)
00:22:33 09.Looking for the Rainbow (Quick ReactorのデフォルトBGM)
00:28:23 10.Speed King (Easy HandlingのデフォルトBGM)
00:32:06 11.Adventure (Wild ChaserのデフォルトBGM)
00:35:53 12.Sonic Control (The Road MonsterのデフォルトBGM)
00:39:46 13.Dream Flying (ドリームフライング)
00:40:47 14.Start (ゲームスタート)
00:42:20 15.Grand Canyon (グランド・キャニオン)
00:43:17 16.San Francisco (サンフランシスコ)
00:44:30 17.South America (南アメリカ)
00:45:24 18.Niagara Falls (ナイアガラの滝)
00:47:05 19.Hawaii (ハワイ)
00:47:45 20.Pacific Ocean Crossway Bridge (太平洋横道橋)
00:48:53 21.Spain (スペイン)
00:50:01 22.Atlantic Ocean Underwater Tunnel (大西洋海底トンネル)
00:51:58 23.Switzerland (スイス)
00:53:09 24.China (中国)
00:54:03 25.Japan (日本)
00:54:54 26.Hong Kong (香港)
00:55:54 27.Egypt (エジプト)
00:56:47 28.Mediterranean Sea (地中海)
00:58:57 29.France (フランス)
01:00:24 30.Germany (ドイツ)
01:01:50 31.Final Stage (最終ステージ)
01:03:21 32.Meaning of the Light (ロングバージョン)
01:05:14 33.Game Over (アイキャッチ)
01:05:18 34.Jingle Bell (ジングルベル)
01:06:25 35.BGM #eb
01:06:56 36.BGM #ee
01:07:36 37.Sound test (サウンドテスト)
01:09:23 38.Meaning of the Light (ゲームオーバーバージョン)
01:10:46 39.Last Wave (1993)
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