The Ideal Formula

	The Ideal Formula; 世界を彩る数式 The Formula Coloring the World
	; Event notification image
Type	Story event
First run	December 24, 2020 to January 12, 2021
Rerun(s)	September 13 to 17, 2021; November 7 to 21, 2022; March 12 to 31, 2025; September 8 to 19, 2025
Featured sync pairs	N & Zekrom
Sync pairs with reward bonus
1.6×	First/third/fourth runs only:; N & Zekrom
1.4×	First/third/fourth runs only:; Hilbert & Oshawott/Dewott/Samurott; Hilda & Tepig/Pignite/EmboarFirst/third runs only:; Elesa & Zebstrika
1.2×	First/third runs only:; Iris & Haxorus; Scottie/Bettie & Pikachu; Scottie/Bettie & Torchic
Theme skills with 1.2× reward bonus	Fourth run only:; Unova

The Ideal Formula (Japanese: 世界を彩る数式 The Formula Coloring the World) was a story event in Pokémon Masters EX. It was focused on the sync pair N & Zekrom.

First run

The event first ran from December 24, 2020, to January 12, 2021. The "N Poké Fair Scout" gave players an increased chance of recruiting N & Zekrom from the beginning of the event to January 29, 2021.

In this run, players could earn Prize Coins from battles that could be used to draw items from a prize box pool. All battles could be challenged an unlimited number of times except Challenge N, which could only be challenged once per day. If the grand prize was drawn from the prize box, players could advance to the next prize pool. The grand prizes included area info for event story areas, Skill Spheres, 5★ Scout Tickets, Power-Up items, a Strike Move Candy Coin, and a Tech Move Candy Coin. Extra Prize Coins were automatically exchanged with coins once the event was finished.

The player could battle using N & Zekrom to obtain +0.6× bonus Prize Coins; Hilbert & Oshawott/Dewott/Samurott, Hilda & Tepig/Pignite/Emboar, or Elesa & Zebstrika to obtain +0.4× bonus Prize Coins; and Iris & Haxorus, Scottie/Bettie & Pikachu, or Scottie/Bettie & Torchic to obtain +0.2× bonus Prize Coins.

This event featured the Full-Force Battle, where players were challenged to defeat a large group of strong opponents. Similar to the Legendary Arena, players could use multiple teams with a limited number of total sync pairs, with the opponents' health carrying over between rounds and with previously used pairs unable to be used again until the challenge is reset. Medals could be earned by completing specific missions.

In addition, players could earn sync orbs for N & Zekrom by completing story areas in the event, and a medal for completing the final story area.

Second run

The event reran from September 13 to 17, 2021 as a part of the Two-Year Anniversary Rerun Event. This run only contained the story areas and the battles between them; the prize box, Prize Coins, Challenge N, and Full-Force Battles did not return. In addition, no missions were available, and no medals could be earned.

Third run

The event reran again from November 7 to 21, 2022. While there was no scout directly corresponding to the event, N & Zekrom were featured in the "Vol. 3: Monthly Poké Fair Scout" which lasted for all of November 2022. A special log-in bonus was available alongside this run, which gave players the song keys for three remixes of songs from Pokémon Black and White:

Song Key No. 05017, which allows Route 10 (Unova) (Pokémon Masters EX Version) to be played in the Jukebox.
Song Key No. 05018, which allows Farewell (Pokémon Masters EX Version) to be played in the Jukebox.
Song Key No. 05019, which allows An Unwavering Heart (Pokémon Masters EX Version) to be played in the Jukebox.

The former two songs were first used in this event, while the third was first used in Father or Foe.

In this run, all the battles and other mechanics from the first run returned, including the prize box, missions, and medals. The grand prizes now included area info for event story areas, Power-Up items, 5★ Unova Scout Tickets, Skill Spheres, and a Support Move Candy Coin. Completing story areas in the event now awarded Co-op Sync Orbs instead of sync-pair-specific sync orbs, and an additional mission was added to award players with a Strike Move Candy Coin for completing the second Full-Force Battle using only three sync pairs.

Fourth run

The event reran again from March 12 to 31, 2025, shortly after the addition of the third chapter of the Arceus arc.

This run followed the same structure of the first and third runs. The grand prizes in the prize box included area info for event story areas and a Support Move Candy Coin. In addition, five new missions were added to allow players to earn 4★ Power-Ups, 5★ Scout Tickets, 5★ Unova Scout Tickets, co-op sync orbs, and a 5★ Power-Up for obtaining Prize Coins.

The player could battle using N & Zekrom to obtain +0.6× bonus Prize Coins; Hilbert & Oshawott/Dewott/Samurott or Hilda & Tepig/Pignite/Emboar to obtain +0.4× bonus Prize Coins; and any unlisted Unova sync pairs to obtain +0.2× bonus Prize Coins.

Fifth run

The event reran again from September 8 to 19, 2025 as 1st place in the Trainer Tales Championship. This run was similarly structured to the second run, but it removed the battles between the story areas as well.

Plot

Cheren is confronting a Masked Man with a Zoroark when he notices a Zekrom flying overhead. They both realize it must belong to N, and the Masked Man takes advantage of Cheren's distraction to escape. N and Zekrom land in a field, where Hilda also spots them. She returns to the Pokémon Center, where she reports to Hilbert what she saw and offers to bring him there. Hilbert accepts her offer and invites the player to join them so he can introduce them to N.

Hilda, Hilbert, Cheren, and the player all return to the field where N landed, but they can't find him anywhere. Hilda asks Hilbert to tell her more about N, and Hilbert explains that N used to be the king of Team Plasma. While the team itself, led by Ghetsis, ultimately had nefarious motives, N genuinely wanted to help Pokémon and thought that separating people and Pokémon was the best way to do that. However, as he traveled around Unova and battled Hilbert, he gradually realized how much people and Pokémon can help each other. All of this has led Hilbert to consider N an important friend. Hilda is filled with new determination help Hilbert find N, so she, Cheren, and Hilbert get back to searching.

Meanwhile, N has arrived in Centra City and is observing sync pairs interacting when Iris and Bianca approach him. Iris is keen to show N around and meet Zekrom, who is flying above the rooftops of the city. This gives Iris an idea, and she leads the group to the Pasio arena. While Bianca goes and gets permission for them to use the arena, Iris starts chatting with N about Pokémon battles. N finds it odd that she and Bianca are being so kind to him, as he thinks they should hate him after everything he did as the king of Team Plasma. Iris says that she knows he was just pursuing his dream of making Pokémon happy, and she could never hate someone for that. Plus, she can tell that Zekrom is comfortable around him. N comments that she doesn't just understand the hearts of dragons, but the hearts of people as well, and he thinks she'll be a good Champion for Unova. Bianca finally returns, having gotten permission to use the arena, and Iris challenges N to a battle. An intense battle follows between Iris's Haxorus and N's Zekrom. After the battle, Bianca notes how late it is and gets ready to leave with Iris, who promises to come back tomorrow and bring Hilbert with her.

As N takes in the news that he's going to see Hilbert again, Hilbert is disappointed that he hasn't been able to find N yet. Hilda wants to keep searching the forest, but Cheren points out how late it is and suggests that they try again tomorrow with Bianca's help. Suddenly, they hear someone cry out. They follow the voice and find Team Break intimidating a Youngster into giving up his Poochyena. When Hilbert, Cheren, Hilda, and the player confront them, the Masked Man with Zoroark agrees to leave the Pokémon alone in exchange for Hilbert and his friends coming with them. When Hilda refuses, the Masked Man reveals that they have already captured the Youngster's little sister. Hilbert agrees to go with them if they let the girl go. With the exchange made, the Team Break members head off to their secret hideout with Hilbert, Hilda, Cheren, and the player in tow.

The next morning, Iris and Bianca meet N on the beach and explain that they haven't been able to get in touch with Hilbert, Hilda, or Cheren. Just then, they're approached by PML staff, who have learned from the Youngster and his sister that Hilbert, Hilda, Cheren, and the player have been kidnapped by Team Break. Iris wants to help them, but she doesn't know where to start looking. However, the Youngster's Poochyena says something to N that he finds interesting.

Meanwhile, Hilda, Hilbert, Cheren, and the player are locked up in Team Break's hideout and wonder what the group wants with them. Frustrated, Hilda demands answers from their captors. The Masked Man reveals that they captured Hilbert to lure N to the hideout and are preparing a "grand welcome" for him. Suddenly, the ground starts shaking, and they get a call from their allies outside the hideout, who are under attack from Bianca and Iris. Team Break is astonished that the girls were able to find their secret hideout, and N reveals that the Youngster's Poochyena remembered their scent and was able to track down their hideout. N declares that he's going to rescue his friends, and the Masked Man with Zoroark comes outside with the hostages. He approaches N, addressing him as "Your Majesty", and reveals that he's a supporter of N's dream to liberate Pokémon from their Trainers. He thinks Pasio is a prison for Pokémon, that Pokémon are trapped on the island the same way humans can trap them inside Poké Balls, and they're being forced to battle each other in the PML. Hilbert, Hilda, Cheren, Bianca, Iris, and the player vehemently protest his claims, but N says nothing. The Masked Man admits that Team Break does awful things, but they have power, so he asks N to become the king of Team Break so they can use Team Break's power to save Pokémon.

Finally, N explains that he has traveled the world and seen people and Pokémon living together in harmony all over, even on Pasio. He's come to understand that people and Pokémon can reach their fullest potential by being together, so the solution is not to separate people and Pokémon, but to bring them together. The Masked Man is dismayed that N has changed, so he declares that he'll just make N's old dream come true himself. The other members of Team Break surround N's group and prepare to battle. Hilda and Cheren take down the other members of Team Break while N and Hilbert battle and defeat Zoroark. The Masked Man surrenders, but he refuses to give up on making N's old dream a reality. However, N points out that his Zoroark fought for him with everything it had because it loves him, even knowing that the his goal would result in them being separated forever. N tells him to listen to his friend, who has so much love for him, and the Masked Man leaves quietly with Zoroark.

That night, Bianca, Iris, Hilda, Cheren, Hilbert, N, and the player gather on the beach. Cheren admits that he has some things he wants to say to N, but he stands back to let him talk to Hilbert first. N says that he's grateful for everything Hilbert taught him, and Hilbert grins and says that that's what friends are for. N is happy to hear Hilbert call him a friend. Then, Hilbert reminds him that just before left to go on his journey, he said that he hoped Hilbert could make his dream come true, and that he believed Hilbert could do it. However, he then said good-bye and left without giving Hilbert the chance to respond, which Hilbert is annoyed by. N laughs and apologizes for his selfishness back then, and he offers to make it up to Hilbert now. Instead of saying good-bye, he says "see you soon!".

Area info

Area name	Unlock condition			Completion rewards
Area name	First/third/fourth runs	Second run	Fifth run	First/third/fourth runs	Second run	Fifth run
The Hero Arrives on Pasio	First run: Plays automatically on log-in Third/fourth runs: Available from start	Available from start		First run: None Third/fourth runs: Gems ×10	Gems ×10 The King of Team Plasma
The King of Team Plasma	Pull from the prize box (Pool 1)	Complete The Hero Arrives on Pasio		Gems ×10	Gems ×10 Challenge Pasio's Trainers: Part 1	Gems ×10 The Ones Who Bested Alder
The Ones Who Bested Alder	Pull from the prize box (Pool 2)	Complete Challenge Pasio's Trainers: Part 1	Complete The King of Team Plasma	Gems ×10	Gems ×10 Challenge Pasio's Trainers: Part 2	Gems ×10 N to the Rescue
N to the Rescue	Pull from the prize box (Pool 3)	Complete Challenge Pasio's Trainers: Part 2	Complete The Ones Who Bested Alder	Gems ×10	Gems ×10 Challenge Pasio's Trainers: Part 3	Gems ×10 A Gray World
A Gray World	Pull from the prize box (Pool 4)	Complete Challenge Pasio's Trainers: Part 3	Complete N to the Rescue	Gems ×10	Gems ×10 Challenge Pasio's Trainers: Part 4	Gems ×10 No More Good-byes
No More Good-byes	Pull from the prize box (Pool 5)	Complete Challenge Pasio's Trainers: Part 4	Complete A Gray World	Gems ×10

Battles

First/third/fourth runs

Name	Difficulty	Trainer	Pokémon			Stamina	First-Time Rewards	Completion Rewards
Name	Difficulty	Trainer	Left	Middle	Right	Stamina	First-Time Rewards	Completion Rewards
Challenge Pasio's Trainers: Part 1	Normal	Collector Norris	Carracosta Weakness:	Kommo-o Weakness:	Crobat Weakness:	10	Gems ×30	200 coins Prize Coins ×12
Challenge Pasio's Trainers: Part 2	Hard					15	Gems ×30	First run: 450 coins Prize Coins ×23 Third/fourth runs: 400 coins Prize Coins ×20
Challenge Pasio's Trainers: Part 3	Very Hard					15	Gems ×30	1,000 coins First run: Prize Coins ×29 Third/fourth runs: Prize Coins ×25
Challenge Pasio's Trainers: Part 4	Super Hard					20	Gems ×30	1,500 coins First run: Prize Coins ×50 Third/fourth runs: Prize Coins ×40
Challenge N	Ultra Hard	N	Pidgeot Weakness:	Zekrom Weakness:	Staraptor Weakness:	30	Gems ×30	2,000 coins First run: Prize Coins ×225 Third/fourth runs: Prize Coins ×150
Full-Force Battle! Round 1	Super Hard	Masked Man	Salamence Weakness: ↓ Garchomp Weakness: ↓ Sharpedo Weakness:	Aerodactyl Weakness: ↓ Kingdra Weakness: ↓ Kabutops Weakness:	Turtonator Weakness: ↓ Haxorus Weakness: ↓ Walrein Weakness:	—	Gems ×100 Full-Force Battle! Round 2 First run: 3★ Tech Move Candy Third/fourth runs: 3★ Support Move Candy	None
Full-Force Battle! Round 2	Ultra Hard	Masked Man				—	Gems ×100 4★ Strike Move Candy	None

Second run

Name	Difficulty	Trainer	Pokémon			First-Time Rewards	Completion Rewards
Name	Difficulty	Trainer	Left	Middle	Right	First-Time Rewards	Completion Rewards
Challenge Pasio's Trainers: Part 1	Normal	Collector Norris	Carracosta Weakness:	Kommo-o Weakness:	Crobat Weakness:	Gems ×100 The Ones Who Bested Alder 4★ Level-Up Manual ×5 3-Pack Great Drink ++ Set ×10 3-Pack Drink ++ Set ×15 3-Pack Ultra Drink + Set ×20 3-Pack Great Drink + Set ×30 3-Pack Drink + Set ×40 3-Pack Ultra Drink Set ×50 3-Pack Great Drink Set ×60	200 coins
Challenge Pasio's Trainers: Part 2	Hard					Gems ×100 N to the Rescue 4★ Level-Up Manual ×5 3-Pack Great Drink ++ Set ×10 3-Pack Drink ++ Set ×15 3-Pack Ultra Drink + Set ×20 3-Pack Great Drink + Set ×30 3-Pack Drink + Set ×40 3-Pack Ultra Drink Set ×50 3-Pack Great Drink Set ×60	450 coins
Challenge Pasio's Trainers: Part 3	Very Hard					Gems ×100 A Gray World 4★ Level-Up Manual ×5 3-Pack Great Drink ++ Set ×10 3-Pack Drink ++ Set ×15 3-Pack Ultra Drink + Set ×20 3-Pack Great Drink + Set ×30 3-Pack Drink + Set ×40 3-Pack Ultra Drink Set ×50 3-Pack Great Drink Set ×60	1,000 coins
Challenge Pasio's Trainers: Part 4	Super Hard					Gems ×100 No More Good-byes 4★ Level-Up Manual ×5 3-Pack Great Drink ++ Set ×10 3-Pack Drink ++ Set ×15 3-Pack Ultra Drink + Set ×20 3-Pack Great Drink + Set ×30 3-Pack Drink + Set ×40 3-Pack Ultra Drink Set ×50 3-Pack Great Drink Set ×60	1,500 coins

Event missions

December 24, 2020 to January 12, 2021

Mission Name	Objective	Reward
Mission 1	Complete the following area ×1 The King of Team Plasma	Sync Orb ×50 N & Zekrom
Mission 2	Complete the following area ×1 The Ones Who Bested Alder	Sync Orb ×50 N & Zekrom
Mission 3	Complete the following area ×1 N to the Rescue	Sync Orb ×50 N & Zekrom
Mission 4	Complete the following area ×1 A Gray World	Sync Orb ×50 N & Zekrom
Mission 5	Complete the following area ×1 No More Good-byes	1★ The Ideal Formula
Mission 6	Complete any of the following areas ×3 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4	Gems ×50
Mission 7	Complete any of the following areas ×10 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4	Gems ×100
Mission 8	Complete any of the following areas ×20 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4	Gems ×100
Mission 9	Complete any of the following areas ×30 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4	Gems ×100
Mission 10	Complete any of the following areas ×40 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4	Gems ×100
Mission 11	Complete any of the following areas ×50 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4	Gems ×100
Mission 12	Complete the following area ×1 Full-Force Battle! Round 1	2★ The Ideal Formula
Mission 13	Complete the following area ×1 Full-Force Battle! Round 2	3★ The Ideal Formula
Mission 14	Complete the following area with 6 or fewer sync pairs falling to 0 HP: Full-Force Battle! Round 2	N-Describable Love
Reward for completing all missions		Gems ×200

November 7 to 21, 2022

Mission Name	Objective	Reward
Mission 1	Complete the following area ×1 The King of Team Plasma	Co-op Sync Orb ×200
Mission 2	Complete the following area ×1 The Ones Who Bested Alder	Co-op Sync Orb ×200
Mission 3	Complete the following area ×1 N to the Rescue	Co-op Sync Orb ×200
Mission 4	Complete the following area ×1 A Gray World	Co-op Sync Orb ×200
Mission 5	Complete the following area ×1 No More Good-byes	1★ The Ideal Formula
Mission 6	Complete any of the following areas ×3 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×50
Mission 7	Complete any of the following areas ×10 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×50
Mission 8	Complete any of the following areas ×20 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×100
Mission 9	Complete any of the following areas ×30 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×100
Mission 10	Complete any of the following areas ×40 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×100
Mission 11	Complete any of the following areas ×50 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×100
Mission 12	Complete the following area ×1 Full-Force Battle! Round 1	2★ The Ideal Formula
Mission 13	Complete the following area ×1 Full-Force Battle! Round 2	3★ The Ideal Formula
Mission 14	Complete the following area with 6 or fewer sync pairs falling to 0 HP: Full-Force Battle! Round 2	N-Describable Love
Mission 15	Complete the following area with 3 or fewer sync pairs falling to 0 HP: Full-Force Battle! Round 2	Strike Move Candy Coin
Reward for completing 14 missions		Gems ×200

March 12 to 31, 2025

Mission Name	Objective	Reward
Mission 1	Complete the following area ×1 The King of Team Plasma	Co-op Sync Orb ×200
Mission 2	Complete the following area ×1 The Ones Who Bested Alder	Co-op Sync Orb ×200
Mission 3	Complete the following area ×1 N to the Rescue	Co-op Sync Orb ×200
Mission 4	Complete the following area ×1 A Gray World	Co-op Sync Orb ×200
Mission 5	Complete the following area ×1 No More Good-byes	1★ The Ideal Formula
Mission 6	Complete any of the following areas ×3 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×50
Mission 7	Complete any of the following areas ×10 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×50
Mission 8	Complete any of the following areas ×20 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×100
Mission 9	Complete any of the following areas ×30 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×100
Mission 10	Complete any of the following areas ×40 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×100
Mission 11	Complete any of the following areas ×50 • Challenge Pasio's Trainers: Part 1 • Challenge Pasio's Trainers: Part 2 • Challenge Pasio's Trainers: Part 3 • Challenge Pasio's Trainers: Part 4 • Challenge N	Gems ×100
Mission 12	Obtain: Prize Coins ×300	4★ Power-Up ×5
Mission 13	Obtain: Prize Coins ×600	5★ Unova Scout Ticket ×8
Mission 14	Obtain: Prize Coins ×1,000	5★ Scout Ticket ×8
Mission 15	Obtain: Prize Coins ×1,500	Co-op Sync Orb ×5,000
Mission 16	Obtain: Prize Coins ×2,000	5★ Power-Up ×1
Mission 17	Complete the following area ×1 Full-Force Battle! Round 1	2★ The Ideal Formula
Mission 18	Complete the following area ×1 Full-Force Battle! Round 2	3★ The Ideal Formula
Mission 19	Complete the following area with 6 or fewer sync pairs falling to 0 HP: Full-Force Battle! Round 2	N-Describable Love
Mission 20	Complete the following area with 3 or fewer sync pairs falling to 0 HP: Full-Force Battle! Round 2	Strike Move Candy Coin
Reward for completing 19 missions		Gems ×200

Prize box

December 24, 2020 to January 12, 2021

Prize pool	Grand prize
1	The King of Team Plasma
2	The Ones Who Bested Alder
3	N to the Rescue
4	A Gray World
5	No More Good-byes
6	5★ Scout Ticket ×2
7	4★ Power-Up ×1
8	5★ Scout Ticket ×2
9	Skill Sphere ×200
10	5★ Scout Ticket ×2
11	Skill Sphere ×200
12	5★ Scout Ticket ×2
13	5★ Power-Up ×1
14	Strike Move Candy Coin ×1
15	5★ Power-Up ×1
16	Tech Move Candy Coin ×1
17	3★ Power-Up ×1

November 7 to 21, 2022

Prize pool	Grand prize
1	The King of Team Plasma
2	The Ones Who Bested Alder
3	N to the Rescue
4	A Gray World
5	No More Good-byes
6	3★ Power-Up ×3
7	5★ Unova Scout Ticket ×2
8	4★ Power-Up ×3
9	5★ Unova Scout Ticket ×2
10	4★ Power-Up ×3
11	5★ Unova Scout Ticket ×2
12	Skill Sphere ×200
13	5★ Scout Ticket ×2
14	Support Move Candy Coin ×1
15	3★ Power-Up ×1

March 12 to 31, 2025

Prize pool	Grand prize
1	The King of Team Plasma
2	The Ones Who Bested Alder
3	N to the Rescue
4	A Gray World
5	No More Good-byes
6	Support Move Candy Coin ×1
7	3★ Power-Up ×1

Gallery


Event banner (First/third/fourth runs)

Event banner (Two-Year Anniversary Rerun)

Medal 1★ The Ideal Formula	Medal 2★ The Ideal Formula	Medal 3★ The Ideal Formula	Medal N-Describable Love	Artwork by mikko

Music

Situation	Song key	Song name	Arrangement
In menu	No. 05017	Route 10 (Unova) (Pokémon Masters EX Version)	Unknown
During battle (Challenge Pasio's Trainers)	No. 00017	Battle! (Sync Pair)	Shota Kageyama
During battle (Challenge N) The Ones Who Bested Alder As N and Iris battle A Gray World When N accepts the Masked Man's challenge No More Good-byes From start	No. 05002	Battle! (N) (Pokémon Masters EX Version)	Haruki Yamada (ATTIC INC.)
During battle (Full-Force Battle) A Gray World When the Masked Man challenges N	No. 00023	Battle! (Team Break)	Haruki Yamada (ATTIC INC.)
The Hero Arrives on Pasio From start	No. 00010	Wild Wanderings	Unknown
The Hero Arrives on Pasio When N first lands on Pasio The King of Team Plasma As Hilbert and Cheren explain to Hilda about N The Ones Who Bested Alder When N questions Iris's welcoming demeanor, and when Hilbert, Hilda, Cheren, and the player find Team Break in the forest N to the Rescue When Iris, Bianca, and N are informed of Hilbert and his friend's whereabouts	No. 00014	Lurking Threat	Unknown
The Hero Arrives on Pasio When Hilda meets Hilbert at the Pokémon Center	—	Pokémon Center (Pokémon Masters EX Version)	Unknown
The King of Team Plasma From start The Ones Who Bested Alder As Hilbert, Hilda, Cheren, and the player continue to search for N N to the Rescue From start	No. 00008	Out for a Stroll	Unknown
The King of Team Plasma When N first enters the city The Ones Who Bested Alder From start	No. 00007	Fun with Friends	Unknown
The Ones Who Bested Alder When Iris brings N to the Pasio arena	No. 00003	Taking It Easy	Unknown
N to the Rescue When Iris, Bianca, and N infiltrate Team Break's hideout	No. 00005	Danger!	Unknown
N to the Rescue When Hilbert finally meets N again A Gray World From start	No. 00016	Sneaking Suspicion	Unknown
A Gray World As N expresses his view on the relationship between people and Pokémon No More Good-byes When N tells the Masked Man to listen to his Pokémon	No. 00013	A Somber Moment	Unknown
No More Good-byes After N and Hilbert defeat the Masked Man	No. 00009	Island Mysteries	Unknown
No More Good-byes As Hilbert and N talk alone	No. 05008	Farewell (Pokémon Masters EX Version)	Unknown
Song names without a song key listed are unofficial, as they have not been added to the Jukebox.

Script

Pokémon Center (event conversation)

Name	Text
Hilbert	After all this time, I can finally see him again, here on Pasio...
Hilbert	There's a lot of stuff I need to say to N! Let's go look for him, <player>!

The Hero Arrives on Pasio

Name	Text
(Zekrom flies over Cheren while battling Team Break)
Cheren	Was that...Zekrom?! Then that means...
Masked Man	No way! HE came to Pasio?!
(Zekrom lands in a field)
???	So, this is where you wanted to bring us.
N	The artificial island of Pasio.
(Hilda looks from afar)
Hilda	Hey, is that...
(Transition to the Pokémon Center)
Hilbert	No way! Are you sure?!
Hilda	I'm positive! You've told me all about him, so I knew exactly who it was! The person I saw was definitely N!
Hilbert	Wow. So N is here on Pasio... I haven't seen him in so long.
Hilda	Hilbert... You really want to see him again, don't you. Why don't we go look for N? I'll show you where I saw him!
Hilbert	Yeah, let's go! Thanks! <player>, why don't you come with us, too? I wanna introduce you to my friend!

The King of Team Plasma

Name	Text
(Field)
Hilda	That's odd—he was just here earlier. I wonder where he went.
Cheren	Well, he never was the type to stay in one place for very long.
Hilbert	If you both saw him, then he was definitely here for sure!
Cheren	Yeah, but Team Break got away while I was distracted. I still have so much to learn.
Hilbert	Hey, I get it. This is N we're talking about, after all.
Hilda	You know, I've been wondering about this for a while now... but who exactly is this N guy, anyway? I mean, what's his deal?
Hilbert	Wait, you don't know, Hilda? Even after everything that happened in Unova?
Hilda	I-I just like to focus on battling—that's all! Besides, <player> also needs to know what's going on, right?
Cheren	Yes, I do!: Yes, that's probably for the best.
Hilbert	Who is N?: Yeah, we should bring you up to speed before we go meet him. This N we're searching for... He used to be the king of Team Plasma.
Hilda	Whoa! Team Plasma?! Even I know about them! They're that group who'd say stuff about how we need to liberate Pokémon from their human oppressors. But in reality, they were just a bunch of Pokémon-stealing jerks!
Hilbert	Yeah, we had a few run-ins with Team Plasma in the past. We also clashed with N a few times.
Cheren	But the real mastermind was Ghetsis. He was controlling everything for his own gain. After Ghetsis was apprehended, Team Plasma laid low for a while.
Hilbert	But N's feelings toward Pokémon—his beliefs...they were genuine.
Hilda	His beliefs?
Hilbert	He believed that Pokémon suffered because of misguided humans... So to change this messed-up world, he thought we needed to separate Pokémon and humans. All to help Pokémon, our friends...
Cheren	Ghetsis exploited N's beliefs, and we ended up battling each other. Looking back, I can't believe it even came to that. Even though we all cared about our Pokémon...we still had to fight.
Hilda	Hilbert, Cheren... N must have a really pure heart, huh?
Hilbert	But through our battles and travels, N's beliefs began to waver. He saw with his own eyes how Pokémon and humans can understand and help each other. And he made a real effort to try to understand how we felt. That's why N is a really important friend to me!
Hilda	A friend... I see! I'm still trying to process all this, but I think I get how you feel. Let's go find your friend so you can see him again!
Hilbert	Thanks!
Cheren	That said, N is really unpredictable. It might take a while to figure out where he is.
Hilbert	No matter how hard it is, we'll definitely find N. There's a lot of stuff I need to say to him!
Cheren	Like what?
Hilbert	Nothing major, but...something important. OK, let's head out and find N!
(Transition to N in town)
N	So this is Pasio... An artificial island where humans and Pokémon can coexist...
???	I knew it was you! Hey there, N!
(Bianca and Iris appear)
N	You two...

The Ones Who Bested Alder

Name	Text
Iris	C'mon, N! Over here!
N	Sure.
Iris	Jeez! Don't be such a Slowpoke! The sun's about to set, and we haven't even finished showing you around town!
Bianca	chuckle There's no rush, Iris. He did just get here, after all! Let us know if anything catches your eye, N. We'd be happy to give you a closer look.
N	Yes...thank you.
Iris	By the way, where's Zekrom? Is it in a Poké Ball right now?
N	No. It seems Zekrom is also very interested in Pasio.
(Zekrom flies over the town)
Bianca	Whoa! So that's Zekrom, huh? It's so cool!
Iris	Wow...a Legendary Dragon Pokémon... OK, I know where we should go next! Follow me!
Bianca	Wait for us, Iris!
N	... ... ...
(Transition to the arena)
Iris	This is the Pasio arena! It's normally used for Pokémon battles, but there aren't any matches happening today.
N	Pokémon battles...
Bianca	Iris, I'll go get us permission to use the arena!
Iris	Oh... Do you not like Pokémon battles, N?
N	No, it's not that...
(Zekrom appears)
Iris	Ah! Zekrom!
N	A Pokémon battle is a way for us to see what wonderful things others have to contribute. Accepting different ideas and different beings changes the world like a chemical reaction. Zekrom taught me that...
Iris	Different ideas? A chemical reaction? I don't really get all that, but I guess it means you don't hate Pokémon battles! Me, I love to battle! Oh? Yes, I know.
N	You people are very strange.
Iris	Hm? How so?
N	You know what I did in Unova, don't you? I was the king of Team Plasma. We fought the Gym Leaders and the Champion. We opposed everything you stood for. You have every reason to hate me. I don't see why you would be kind to me. So...why? Why are you being so welcoming?
Iris	Why not?
N	What do you mean?
Iris	You know, the former Champion of Unova, Alder, was really, really strong. But he lost to you, right?
N	... ... ...
Iris	Alder always used to say... the ones who change the world are always the ones who pursued their dreams. I think that's why you beat him! Because you've been pursuing your dream of making Pokémon happy!
N	Wha—
Iris	There's no way I could hate someone like that! And also... Zekrom is so calm around you. I know you can understand what Pokémon say, but I can tell what dragons are feeling!
N	... ... ... You're different from me, Iris.
Iris	Huh?
N	It's not just that you can understand the hearts of dragons. You also understand the hearts of people. With you as Champion, Unova will become a place where Pokémon can live peacefully.
Iris	N... Thanks! I'll do my best!
Bianca	Yoo-hoo, Iris! I got permission to use the arena!
Iris	Thanks, Bianca! N... Battle me! I've always wanted to battle you! You defeated Alder, after all!
N	Iris... Yes, I'd like that, too. I want to see the power of the current Unova Champion! I'll surpass you!
(Iris and N battle)
(Transition to sunset)
Iris	Phew... Boy, are you two strong!
N	You're strong, too, Iris. I wonder what would have happened if you had been the Champion I challenged back then.
Bianca	It's getting late. Let's head out, shall we?
Iris	Let's hang out again tomorrow! I'll bring Hilbert along, too!
N	Hilbert?! He's on Pasio, too?
Bianca	Come to think of it, he's been wanting to see you again for a while now! We probably should have invited him today.
Iris	Then I'll definitely bring him with me tomorrow!
N	I'm going to see him again...
(Transition to the forest)
Cheren	The sun's starting to set... Guess we couldn't find any clues to N's whereabouts.
Hilda	Let's keep searching for him.: Yeah! It's still too early to give up!
Cheren	Let's keep searching for him.: I understand how you both feel, but it's getting dark. Let's stop for today. Let's try again tomorrow.: Good idea. It'll be harder to find him when it's dark out, anyway.
Hilbert	Yeah. What a bummer, though.
Hilda	Hey, cheer up! We know he's here somewhere!
Cheren	Let's ask Bianca to help us tomorrow. I'm sure we'll find him if we do.
Hilbert	Good idea.
???	NOOO! Go away!
Cheren	What was that?!
Hilbert	It came from over there!
(Team Break attacking a Youngster)
Masked Man	Give us your Pokémon! It's for its own good!
Youngster	No! Poochyena's my friend!
Masked Man	Your "friend," huh? My Zoroark is strong! I've spent years raising it!
Hilbert	Hey, cut it out!
Cheren	If you're going to resort to force, you'll have to go through us.
Masked Man	Who are you? Hm? You...
Hilbert	Well? You gonna take us on or what?
Masked Man	Very well... We won't take this Pokémon. However! You're coming with us to our secret hideout!
Hilda	What in the world are you talking about? There's no way we're going with you.
Masked Man	Heh heh heh... Maybe this'll change your mind.
(Team Break holding a Preschooler hostage)
Preschooler	Waaah! Waaah! I want my big brother!
Hilda	That little girl!
Cheren	Tch! What a dirty trick.
Masked Man	Well? What're you gonna do?
Hilbert	Fine, let's go.
Masked Man	Heh heh... How very cooperative of you!
(The group leaves with Team Break)
Youngster	What do we do now?

N to the Rescue

Name	Text
(Beach)
N	This is a world created by humans...
Bianca	Heeeey, N!
N	Good morning, Bianca, Iris. Oh? Hilbert's not with you?
Iris	It's really weird—for some reason, we can't get ahold of him!
Bianca	And not just him! Cheren and Hilda, too! It's not like them at all. I hope nothing bad has happened...
???	Pardon me!
(A Pokémon Ranger appears with the Preschooler and Youngster)
Pokémon Ranger	I'm with PML staff. Do you have a moment to answer a few questions?
(Screen fades to black, then back)
Bianca	That's them! It's Hilbert and our friends!
Iris	They were kidnapped by Team Break?!
Youngster	They did it to save us...
Preschooler	Waaaah! I'm soweeee!
Bianca	I-it's OK. Don't cry. Hilbert and the others will be OK.
Iris	I wanna go help them...but we don't know where they went...
(The Youngster's Poochyena barks)
N	Really?
Bianca	What is it, N?
N	... ... ...
(Transition to the player's group in the Team Break hideout)
Cheren	It's been a while since they brought us here, but we still have no idea what Team Break is up to.
Hilbert	Yeah. I can't believe they're just keeping us locked up here.
Cheren	What in the world are they thinking?: No idea. That's why it's so creepy.
Hilbert	What if they never let us go?: No way! We'll find a way out of here—don't you worry!
Cheren	Focus on staying calm. That way, we can be prepared to deal with whatever they try to do.
Hilda	Argh! I can't take it anymore! Hey! Team Break! Come out and face us!
Hilbert	Shhh, Hilda!
Masked Man (1)	Well, aren't you a feisty one? Sorry, but we're in the middle of preparing a grand welcome for N.
Hilbert	N?! What's he got to do with this?
Masked Man (1)	If you're here, N will surely come. You're the reason Team Plasma fell from grace... I'm just using you so I can achieve my dream!
(Tremor)
Masked Man (1)	What's going on?! Why is the ground shaking?!
Masked Man (2)	Sir, we have a problem!
(Transition to Bianca and Iris battling Team Break)
Iris	Go get 'em, Haxorus!
Bianca	Help us out, Musharna!
Masked Man (3)	How did they find our secret hideout?!
???	This little one remembered your scent and led us right to you.
(N appears with the Youngster and his Poochyena)
N	I'm here to rescue my friends! No matter what!
Hilbert	N!
N	Hilbert...
Bianca	Are you all OK?!
Iris	Hold tight! We'll get you out of this!
Masked Man (1)	I-it's really you... I've dreamed of meeting you, N. Your Majesty!
Iris	Wha—?
N	... ... ...

A Gray World

Name	Text
(Team Break lets go of the player's group)
Hilda	I'm glad we're getting out of this mess, but...
Hilbert	N, do you know this guy?
N	No. At least...I don't think so...
Masked Man	Of course you wouldn't know me. I'm not from Unova. But I believe wholeheartedly in your dream!
Cheren	N's dream?
Masked Man	Have you had a chance to walk the streets of Pasio, Your Majesty?
N	Yes.
Masked Man	Wasn't it just awful? "A paradise where humans and Pokémon coexist." That's supposedly the concept of this island. But nothing could be further from the truth! This place is nothing but a glorified prison for Pokémon...
N	Pasio is...a prison?
Hilda	That's nonsense! You don't know what you're talking about!
Masked Man	Nonsense, you say? Think about it! Pokémon are trapped on this island and have their freedom taken away! It's no different from the oppression of a Poké Ball! And they're forced to battle each other in that ridiculous PML tournament! Even Pokémon who don't want to battle are forced to do so against their will! This island is nothing but a false utopia, created by egotistical humans!
N	... ... ...
Bianca	That's not what Pasio is!
Cheren	And where do you get off lecturing us after all the bad things Team Break has done?
Masked Man	Yes, I agree. Team Break is the worst. They selfishly use Pokémon for their own desires. However, they do have power. And I wanted to use that power to save Pokémon! But I wasn't strong enough... Your Majesty! Please, help me save the Pokémon! Become the king of Team Break!
Iris	The king of Team Break?!
Hilbert	N...
Masked Man	We're nothing like Ghetsis! We really want to save Pokémon! We want to change this gray world, where black and white—people and Pokémon—have been mixed together instead of kept separate... So please! Lend us your strength!
N	... ... ...
Masked Man	Your Majesty!
N	I have traveled the world and seen much of it... No matter where I went, the relationship between people and Pokémon was the same. What did you call it? Ah yes, a gray world.
Masked Man	Exactly! That's what we need to change!
N	However... That's how you see the world. It's not what I see.
Masked Man	Wha—
N	This world isn't gray at all. You only see it that way because you're not trying to see the truth! I've seen with my own eyes people and Pokémon coexisting. I saw it here on this paradise island called Pasio!
Masked Man	Your Majesty...
N	Pokémon can unleash their true power by being with people. And people can discover their true potential by being with Pokémon. Instead of separating black and white, we must accept it. That's what I learned from my travels around the world! The formula that makes up this world!
Hilbert	N...
Masked Man	Your Majesty, no... It can't be... Even if Team Plasma has changed, I didn't want to believe that you've changed, too. Fine. I'll do it myself. I will be the successor of your old dream! Right after I take you down!
(Team Break reinforcements appear)
Cheren	After all that talk, you're just going to resort to brute force?
Hilda	We'll handle these ones back here! N, Hilbert—you guys take him down!
Hilbert	Yeah! C'mon, N!
N	Yeah! We'll protect the dreams of everyone here!

No More Good-byes

Name	Text
(Bianca, Cheren, Hilda, Iris, and the player battle and defeat Team Break)
Hilbert	Now's your chance, N!
N	Let's do it, Hilbert!
(Hilbert and N battle and defeat the masked man)
Masked Man	Argh! Why are you so strong?
Hilbert	It's over. Just surrender already.
Masked Man	I don't want to, but...fine. There's no way I can win here. But I haven't given up! I'll be back to make your old dream a reality!
N	... ... ...
Hilbert	N!
N	This is why we shouldn't separate people and Pokémon.
Masked Man	What?!
N	Your friend loves you. It says it wants to be with you forever. That's why it battled us with everything it had. If you make your dream a reality, you'll be separated forever. And yet, your friend still fought its hardest for you...
Masked Man	Zoroark...
N	Instead of listening to the things I said in the past, maybe you should listen more to your friend here. Your Pokémon has so much love for you.
Masked Man	... ... ... Let's go, Zoroark.
(Transition to the beach at night)
Iris	I was really worried when I heard you'd been kidnapped, but I'm so glad you're all OK!
Bianca	Yeah! And we finally were able to get Hilbert and N together.
Cheren	Yes, I'm glad, too.
Hilda	Hm? What's wrong, Cheren?
Cheren	Did you wanna talk to N, too?: Yeah. I have my own history with him. Now that I'm a Gym Leader, I think I'd be able to approach our conversation from a different point of view. Are you worried about Hilbert?: Yeah. I'm just wondering what was so important that he had to tell N in person. I'd like to talk to N, too... but there's a proper order to this sort of thing, you know? So I'll let Hilbert go first.
N	Bianca told me you've been looking for me.
Hilbert	Yeah...
N	I thought about you a lot during my travels. Our run-ins, our battles, and... what you taught me. I wouldn't be the person I am today if it weren't for you.
Hilbert	N... Don't even sweat it! That's what friends are for, you know?
N	chuckle True... We are friends.
Hilbert	Speaking of which, there's something I've been wanting to talk to you about!
N	What is it?
Hilbert	Before you left to go on your journey... you told me I should make my dream come true—and that you believed I could do it... Then you just up and left after saying good-bye! I mean, what gives?! Give me a chance to respond, at least!
N	... ... ... Pfft! Ahahahaha! You've been looking for me all this time just to tell me that?
Hilbert	D-don't laugh! Maybe it's a little childish, but...
N	No, I'm sorry. That really was selfish of me, wasn't it. Let me make it up to you. I plan on staying on Pasio for a while, anyway. Instead of good-bye this time, maybe I should say... see you soon!

Trivia

The Ideal Formula was the first story event to receive three different runs.
- It was also the first story event to receive four different runs.
The Ideal Formula is the only story event to use a song other than the standard "Adventure Awaits!" as its menu music.
Upon the conclusion of the final story area No More Good-byes, a shortened remix of the ending theme of Pokémon Black and White, Ending "Onward to Our Own Futures", is played.
Since version 2.43.0, released on March 27, 2024, The Ideal Formula has been included as one of the tips that are displayed while downloading data from new app updates. The tip features the artwork of N, Hilbert, and Zekrom released to commemorate the event and a quote from N in the story area The Ones Who Bested Alder: "I'm going to see him again...".

In other languages


Language		Title
Japanese		世界を彩る数式
Mandarin Chinese		世界的理想公式
French		L'équation parfaite
German		Die Formel für eine ideale Welt
Italian		La formula che dà colore
Korean		세계를 색칠하는 수식
Spanish		La fórmula para colorear

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The Ideal Formula

First run

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December 24, 2020 to January 12, 2021

November 7 to 21, 2022

March 12 to 31, 2025

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December 24, 2020 to January 12, 2021

November 7 to 21, 2022

March 12 to 31, 2025

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In other languages

See also

Sync pairs with reward bonus
The Ideal Formula 世界を彩る数式 The Formula Coloring the World
Event notification image
Type	Story event
First run	December 24, 2020 to January 12, 2021
Rerun(s)	September 13 to 17, 2021 November 7 to 21, 2022 March 12 to 31, 2025 September 8 to 19, 2025
Featured sync pairs	N & Zekrom
1.6×	First/third/fourth runs only: N & Zekrom
1.4×	First/third/fourth runs only: Hilbert & Oshawott/Dewott/Samurott Hilda & Tepig/Pignite/Emboar First/third runs only: Elesa & Zebstrika
1.2×	First/third runs only: Iris & Haxorus Scottie/Bettie & Pikachu Scottie/Bettie & Torchic
Theme skills with 1.2× reward bonus	Fourth run only: Unova