Update: The contest goes live today, 4th March 2024 at 9 PM IST.

Warm greetings!

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 4th March 2024 at 9 PM IST.

Registration Link: CodeRush

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all.

The problems are prepared by spongebobsquareroot.

We would also like to thank pradumangoyal and gkapatia for co-ordinating the round.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants who solve at-least 1 problem.

Have an amazing contest! See you all at the leaderboard! :)

Update: The contest goes live today at 9pm.

Warm greetings and Happy New Year everyone!

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 29th January 2024 at 9 PM IST.

Registration Link: CodeRush

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all.

The problems are prepared by spongebobsquareroot.

We would also like to thank pradumangoyal and gkapatia for co-ordinating the round.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants who solve at-least 1 problem.

Have an amazing contest! See you all at the leaderboard! :)

Warm greetings and Merry Christmas everyone,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 27th December 2023 at 9 PM IST.

Registration Link: CodeRush

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems are written by spongebobsquareroot.

We would also like to thank pradumangoyal and gkapatia for co-ordinating the round.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: There might be a slight delay in processing of international prizes.

Have an amazing festive season! See you all at the leaderboard! :)

Update: The contest has been shifted to 12th April 2023, 9 PM IST to stabilise the revamps to the platform architecture. Apologies for the inconveniences caused.

Update: The contest has been shifted to 4th February 2023, 9 PM IST (different problem set). We sincerely apologize for the inconvenience caused by the recent contest crash. We understand that many of our valued participants were affected by this issue and we deeply regret any frustration or disappointment this may have caused. Rest assured that we are taking all necessary steps to prevent such an incident from happening in the future and to ensure that all participants have a fair and enjoyable experience. Thank you for your understanding and patience.

Warm greetings,

Newton School cordially invites you to be a part of our CodeRush-X. The challenge will go live on 28th January 2023 at 9 PM IST. Do register for the contest and stand a chance to win cash prizes worth ₹1,000,000 and exciting goodies like AirPods 3rd Gen, Amazon Echo 4th Gen, boAt Airdopes 131, ergonomic keyboards, Newton School exclusive merchandise (T-shirts, notebooks, stickers, etc.)

Registration Link: CodeRush-X — we have already crossed 100,000 registrations.

You will be given 8 problems and 180 minutes to solve them. The contest will be rated for all! The initial problems are highly beginner friendly and the text and video editorials will be released post the coding contest.

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, Xzirium, and _Enigma__. We would also like to thank pradumangoyal and gkapatia for co-ordinating the contest.

Contest Highlights

Prizes for top global participants

• The first prize winner gets ₹100,000
• Second Prize — ₹50,000
• Third Prize — ₹25,000
• Fourth Prize — ₹15,000
• Fifth Prize — ₹10,000

Gift Vouchers

• Gift Vouchers worth ₹500 for Top 500 coders.
• Gift Vouchers worth ₹100 for coders ranked between 501 — 2000.
• Gift vouchers worth ₹100 for coders ranked 2001 and above — random 1000 people solving at least 1 problem

Prizes for top Indian participants (if not in global top 5)

• First Prize — ₹10,000 with trophy
• Second Prize — ₹5,000 with trophy
• Third Prize — ₹2,500 with trophy
• Fourth Prize — ₹1,500 with trophy
• Fifth Prize — ₹1,000 with trophy

Prizes for top 3 Indian girl participants (over and above all other prizes)

• First Prize — ₹10,000 with trophy
• Second Prize — ₹5,000 with trophy
• Third Prize — ₹2,500 with trophy

Top 3 coders from Indian colleges with 100+ active participants

• First Prize — ₹500
• Second Prize — ₹200
• Third Prize — ₹100

Goodies for top Indian Coders!

• Top 2: AirPods 3rd Gen
• Ranks 3-10: Amazon Echo 4th Gen
• Rank 11-25: boAt Airdopes 131
• Rank 26-100: Cool sipper
• Fastest Indian solvers for every question: Ergonomic Keyboard

Top 100 Indian Participants: Exclusive Newton School T-shirts and goodies!

We hope you like the contest! Hope to see you all at the leaderboard! :)

Warm greetings,

Newton School cordially invites you to be a part of our 3 year anniversary contest. The challenge will go live on 30th November 2022 at 9 PM IST.

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, Xzirium, and _Enigma__. We would also like to thank pradumangoyal and gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹20,001
   • Second Prize: ₹10,002
   • Third Prize: ₹5,001
   • Fourth Prize: ₹3,003
   • Fifth Prize: ₹2,001
2. ₹333 gift vouchers to the top 50 participants.
3. ₹300 gift vouchers to participants with ranks 3, 6, 9, 12, ..., 333 (all multiples of 3 till 333).
4. Top 999 students get INR 20001 off on all Newton School courses!.
5. Placement opportunities with the top product companies for Indian participants.

See you all at the leaderboard! Happy coding :)

Edit: The editorials have been published.

Warm greetings and Happy Diwali everyone,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 27th October 2022 at 9 PM IST. The contest will be themed around Diwali, the Indian Festival of Lights!

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, Xzirium, and _Enigma__.

We would also like to thank pradumangoyal and gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive the prize money through Paypal. All the other gift vouchers will be sent in INR. There might be a slight delay in processing of international prizes.

We hope you all shine bright in the contest! See you all at the leaderboard! :)

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 28th September 2022 at 9 PM IST.

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, Xzirium, and _Enigma__.

We would also like to thank pradumangoyal and gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive the prize money through Paypal. All the other gift vouchers will be sent in INR.

We hope you like the contest! See you all at the leaderboard! :)

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 31st August 2022 at 9 PM IST.

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, Xzirium, and _Enigma__.

We would also like to thank pradumangoyal and gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.
4. Dragon eggs to random participants (undisclosed).

Note: Top 5 participants from other countries can opt to receive the prize money through Paypal. All the other gift vouchers will be sent in INR.

We hope you like the contest! See you all at the leaderboard! Winter is coming :)

Edit: The editorials for the contest have been published.

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 29th July 2022 at 9 PM IST.

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, Xzirium, and _Enigma__.

We would also like to thank for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive the prize money through Paypal. All the other gift vouchers will be sent in INR.

We hope you like the contest! See you all at the leaderboard!

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 24th June 2022 at 9 PM IST.

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, Xzirium, and _Enigma__.

We would also like to thank gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive the prize money through Paypal. All the other gift vouchers will be sent in INR.

We hope you like the contest! See you all at the leaderboard! :)

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 27th May 2022 at 9 PM IST.

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, Xzirium, and _Enigma__.

We would also like to thank gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive the prize money through Paypal. All the other gift vouchers will be sent in INR.

We hope you like the contest! See you all at the leaderboard! :)

Edit: The prizes have been distributed and the editorials have been published.

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 29th April 2022 at 9 PM IST.

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, and Xzirium.

We would also like to thank gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive an Amazon Gift voucher in their respective currencies. All the other gift vouchers will be sent in INR.

We hope you like the contest! See you all at the leaderboard! :)

Edit: The problems have been opened for practice, the editorials have been published!

Warm greetings,

Newton School cordially invites you to be a part of our Newton's Grand Coding Contest. The challenge will go live on 31st March 2022 at 9 PM IST. Do register for the contest and stand a chance to win cash prizes worth ₹1,000,000 and exciting goodies like AirPods 3rd Gen, Amazon Echo 4th Gen, boAt Airdopes 131, ergonomic keyboards, Newton School exclusive merchandise (T-shirts, sippers, notebooks, stickers, etc.)

Registration Link: Newton's Grand Coding Contest — we have already caused 100,000 registrations.

You will be given 8 problems and 180 minutes to solve them. The contest will be rated for all! The initial problems are highly beginner friendly and the text and video editorials will be released post the grand coding contest.

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, and Xzirium.

We would also like to thank gkapatia for co-ordinating the contest.

Contest Highlights

Prizes for top global participants

• The first prize winner gets ₹100,000
• Second Prize — ₹50,000
• Third Prize — ₹25,000
• Fourth Prize — ₹15,000
• Fifth Prize — ₹10,000

Gift Vouchers

• Amazon Gift Vouchers worth ₹500 for Top 500 coders.
• Amazon Gift Vouchers worth ₹100 for coders ranked between 501 — 2000.
• Amazon gift vouchers worth ₹100 for coders ranked 2001 and above — random 1000 people solving at least 1 problem

Prizes for top Indian participants (if not in global top 5)

• First Prize — ₹10,000 with trophy
• Second Prize — ₹5,000 with trophy
• Third Prize — ₹2,500 with trophy
• Fourth Prize — ₹1,500 with trophy
• Fifth Prize — ₹1,000 with trophy

Prizes for top 3 Indian girl participants (over and above all other prizes)

• First Prize — ₹10,000 with trophy
• Second Prize — ₹5,000 with trophy
• Third Prize — ₹2,500 with trophy

Top 3 coders from Indian colleges with 100+ active participants

• First Prize — ₹500
• Second Prize — ₹200
• Third Prize — ₹100

Goodies for top Indian Coders!

• Top 2: AirPods 3rd Gen
• Ranks 3-10: Amazon Echo 4th Gen
• Rank 11-25: boAt Airdopes 131
• Rank 26-100: Cool sipper
• Fastest Indian solvers for every question: Ergonomic Keyboard

Top 100 Indian Participants: Exclusive Newton School T-shirts and goodies!

Note: Top 5 participants from other countries can opt to receive an Amazon Gift voucher in their respective currencies or through a Paypal transaction. All the other gift vouchers will be sent in INR.

We hope you like the contest! Hope to see you all at the leaderboard! :)

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 28th January 2022 at 9 PM IST. Also, do register for Newton's Grand Coding Contest 2022 and stand a chance to win prizes upto ₹1,000,000.

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by _Enigma__, ShlokG, and iLLusio.

We would also like to thank gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive an Amazon Gift voucher in their respective currencies. All other gift vouchers will be sent in INR.

We hope you like the contest! Hope to see you all at the leaderboard! :)

Edit: The problems have been opened for practice and the editorials have been published.

Congratulations to the top participants:

1. Gennady Korotkevich — tourist
2. Ramazan Rakhmatullin — never_giveup
3. Kalash Gupta — kal013
4. Rahim Mammadli — Nots0fast
5. Anay Karnik — AwakeAnay

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 28th December 2021 at 9 PM IST.

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, and Xzirium.

We would also like to thank gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive an Amazon Gift voucher in their respective currencies or through a Paypal transaction. All the other gift vouchers will be sent in INR.

We hope you like the contest! Hope to see you all at the leaderboard! :)

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 30th November 2021 at 9 PM IST. Registration Link: Newton's Coding Challenge

You will be given 7 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by _Enigma__, ShlokG, and iLLusio.

We would also like to thank gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive an Amazon Gift voucher in their respective currencies. All other gift vouchers will be sent in INR.

We hope you like the contest! Hope to see you all at the leaderboard! :)

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 29th October 2021 at 9 PM IST. Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by _Enigma__, ShlokG, and iLLusio.

We would also like to thank gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive an Amazon Gift voucher in their respective currencies. All other gift vouchers will be sent in INR.

We hope you like the contest! Hope to see you all at the leaderboard! :)

Edit: The problems have been opened for practice, the editorials have been published!

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 30th September 2021 at 9 PM IST. We sincerely apologize for the failed servers in the last round and believe that the same problem won't arise again.

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them. The contest will be rated for all!

The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, Xzirium, and swapnil07.

We would also like to thank gkapatia for co-ordinating the contest.

Highlights of contest:

1. The Prize Money for the top 5 performers are as follows:
   • First Prize: ₹10,000
   • Second Prize: ₹5,000
   • Third Prize: ₹2,500
   • Fourth Prize: ₹1,500
   • Fifth Prize: ₹1,000
2. ₹100 Amazon gift vouchers to the top 50 participants.
3. ₹100 Amazon gift vouchers to 50 randomly selected participants ranked between 51-500.

Note: Top 5 participants from other countries can opt to receive an Amazon Gift voucher in their respective currencies. All other gift vouchers will be sent in INR.

We hope you like the contest! Hope to see you all at the leaderboard! :)

Edit: The registrations closes at 30th September 2021 at 9 PM IST. Do register yourself!

Edit: The gift vouchers have been sent to all the participants. The top 5 participants will receive an additional mail with their prizes soon.

Edit: The prize winners have been announced on the Newton School Contest Page. Please click on the star icon next to contest details for the information corresponding to the winners. The gift vouchers will be sent to the prize winners by Friday next week (17.09.2021).

Edit: The editorials have been published here.

Hello Geeks!!!

Assemble to compete with India’s coding community like never before! Newton School’s Grand Coding Contest will be based on the Avengers theme, so it will not only be fun but plotful too!

Contest link — https://bit.ly/3Dau2ZI
Date — 27 August 2021 (Friday)
Time — 9 PM
Duration — 2.5 hours

You will be given 8 problems and 150 minutes to solve them. The contest will be rated for all!
The problems were written and tested by dnshgyl21, _deactivated_, Sawarnik, _Enigma__, and swapnil07.

Highlights of the contest

Prizes

Gift Vouchers

New Special Prizes

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 30th June 2021 at 9 PM IST. The contest will have our dearest characters from FRIENDS!

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them.

The problems are written and prepared by aniket9465 and swapnil07.

We would also like to thank:

Update: Due a clash with CF round, the contest start time has been moved to 10pm

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 28th May 2021 at 10 PM IST. The contest will be based on the storyline of the Game of Thrones, so don't miss the fun!

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them.

The problems are written and prepared by aniket9465 and swapnil07.

We would also like to thank:

Warm greetings,

Newton School cordially invites you to be a part of our monthly coding contest. The challenge will go live on 26th March 2021 at 9 PM IST. The contest will be based on the storyline of the Harry Potter series, so it'll be fun!

Registration Link: Newton's Coding Challenge

You will be given 6 problems and 150 minutes to solve them.

The problems were written and prepared by aniket9465, jiraya_, and swapnil07.

We would also like to thank:

I have always been fascinated by the idea of random walk. Recently, jiraya_ and I observed some new ideas in this domain. Thus, I wanted to share the same with you and get your insights regarding the same.

Assumptions

1. The initial position of the person is assumed to be $$$X$$$ (positive integer) on the number line
2. The walk is not absolutely random, i.e., the probability of person moving from position $$$Y$$$ to position $$$Y+1$$$ is $$$p$$$, and the probability of person moving from position $$$Y$$$ to position $$$Y-1$$$ is $$$q$$$. Obviously, $$$p + q = 1$$$.
3. There is a pit at position 0, therefore a person who reaches the position 0 does not move further.

Pre-requisites

The proof regarding the pre-requisites used in this blog can be found here.

I have used the following results for this blog:

1. The probability of person to reach position $$$Y=0$$$ from position $$$Y=1$$$ after infinite number of moves is 1 if $$$p<=q$$$, and $$$q / p$$$ if $$$p>q$$$.
2. The expected number of moves for a person to reach position $$$Y=0$$$ from position $$$Y=1$$$ is $$$1/(q-p)$$$ if $$$p <= q$$$.

In this blog, we only consider the scenario for various cases after very large (or maybe infinite) number of moves.

Case 1: $$$p=q$$$

The expected position, $$$E[X]$$$, is X and the expected change in position in a move, $$$E[\Delta X]$$$, is 0.

In each move we will disintegrate the person at position $$$Y$$$ into two halves, each having a probability weight as half the value of the probability weight of the original person. A person at position $$$Y$$$ will then represent half a person at position $$$Y+1$$$, and half a person at position $$$Y-1$$$. Therefore, the expected position of person after each move will be $$$\sum_{i=0}^\infty W_i * i$$$. Initially, $$$W_i=1$$$ for $$$i=X$$$ and 0 otherwise.

Now, let's assume a person is at position $$$Y > 0$$$. Therefore, his current contribution to the expected position is $$$W_Y*Y$$$. After a move, his contribution will change to $$$W_Y/2*(Y+1) + W_Y/2*(Y-1) = W_Y*Y$$$. Therefore, his contribution to the expected position does not change if $$$Y>1$$$. If $$$Y=0$$$, the current contribution of the person to the expected position is 0. After a move, the person does not change his current position, and his contribution does not change as well.

Since, the contribution to the expected value does not change for a person at any position, the expected position remains the same after any number of moves.

Special Note : Even though a person at any position $$$X$$$ attains the position $$$X=0$$$ in a random walk if $$$p=q$$$ with probability 1, the expected position of the person always remains $$$X$$$.

Case 2: $$$p<q$$$

The expected position, $$$E[X]$$$, is 0 and the expected change in position in a move, $$$E[\Delta X]$$$, is 0.

For $$$X = 1$$$, the expected number of moves to reach $$$X=0$$$ is $$$1/(q-p)$$$, and for a value of $$$X$$$ greater than 1, the expected number of moves to reach to position $$$X=0$$$ is $$$X/(q-p)$$$ (using independence or the linearity of expectation).

Let's assume a person is at position $$$Y > 0$$$. After a move, his position will change to $$$p*(Y+1)+q*(Y-1)$$$ that is equal to $$$Y+(p-q)$$$. The decrease in expected position of the person is $$$q-p$$$. So we conclude that whenever a person is at a position $$$Y > 0$$$ on the number line, he will change his expected position by $$$-(q-p)$$$ in one move. We already know the expected number of moves to reach position 0 from position $$$X$$$ is $$$X/(q-p)$$$, Therefore, total decrease in value is $$$(X/(q-p))*(-(q-p))=-X$$$. Therefore, the position of the person after infinite number of moves is 0.

Case 3: $$$p>q$$$

The expected position, $$$E[X]$$$, will tend to infinity and the expected change in position in a move, $$$E[\Delta X]$$$, will tend to $$$(1-(q/p)^X)*(p-q)$$$.

We will use the concept of disintegration again. Let's assume a person is at position $$$Y > 0$$$. Therefore, his current contribution to the expected position is $$$W_Y*Y$$$. After a move, his contribution will change to $$$W_Y*p*(Y+1)+W_Y*q*(Y-1)$$$ that is equal to $$$W_Y*Y+W_Y*(p-q)$$$. Therefore, the increase in expected value for a person at position $$$Y>0$$$ is $$$W_Y*(p-q)$$$.

If $$$Y=0$$$, the current contribution of the person to the expected position is 0. After a move, the person does not change his current position, and his contribution does not change as well. Therefore, if a disintegrated portion of the person reaches the position $$$Y=0$$$, it will no longer contribute to the increase in expected value.

From the pre-requisites, we know the probability that a person at position $$$X=1$$$ will ever reach the position $$$X=0$$$ is $$$q/p$$$. Using independence and Markov Chains, we know the probability that a person at a position $$$X>=1$$$ will ever reach $$$X=0$$$ is $$$(q/p)^X$$$. From this, we conclude that the portion of the person that always stay on the number line with position $$$Y>0$$$, irrespective of the number of moves, is $$$(1-(q/p)^X)$$$. Therefore, the expected increment in value will tend to $$$(1-(q/p)^X)*(p-q)$$$ as the number of moves tend to infinity. Since the minimum increment in expected position after each move is a finite value, the expected position will tend to infinity as the number of moves tend to infinity.

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