Ananya Raga, Bhaskar Tala, Charu Veena, and Devendra Sur are four musicians. Each of them started and completed their training as students under each of three Gurus — Pandit Meghnath, Ustad Samiran, and Acharya Raghunath between 2013 and 2024, including both the years. Each Guru trains any student for consecutive years only, for a span of 2, 3, or 4 years, with each Guru having a different span. During some of these years, a student may not have trained under these Gurus; however, they never trained under multiple Gurus in the same year. In none of these years, any of these Gurus trained more than two of these students at the same time. When two students train under the same Guru at the same time, they are referred to as Gurubhai, irrespective of their gender.

The following additional facts are known.

1. Ustad Samiran never trained more than one of these students in the same year.

2. Acharya Raghunath did not train any of these students during 2015-2018, as well as during 2021-24.

3. Ananya and Devendra were never Gurubhai; neither were Bhaskar and Charu. All other pairs of musicians were Gurubhai for exactly 2 years. 4. In 2013, Ananya and Bhaskar started their trainings under Pandit Meghnath and under Ustad Samiran, respectively

In which year did Charu begin her training under Pandit Meghnath?

The correct answer is 2015.

To determine the exact starting year of Charu's training under Pandit Meghnath, we can logically break down the puzzle step-by-step using the given constraints:

Step 1: Determine the training spans for each Guru
We know that the three Gurus (Pandit Meghnath, Ustad Samiran, and Acharya Raghunath) each train students for a distinct span of 2, 3, or 4 consecutive years.
According to Fact 2, Acharya Raghunath (AR) did not train anyone during 2015-2018 or 2021-2024. Therefore, the only available continuous chronological blocks for AR are 2013-2014 (2 years) and 2019-2020 (2 years). Because a student must train for consecutive years, AR cannot possibly accommodate a 3 or 4-year span in these restricted windows. Thus, Acharya Raghunath's training span must be exactly 2 years.
Consequently, Pandit Meghnath (PM) and Ustad Samiran (US) must have the remaining 3-year and 4-year spans.

Step 2: Analyze the "Gurubhai" pairings
Students are considered Gurubhai if they train under the same Guru at the same time.
Fact 1 states Ustad Samiran never trains more than one student at a time, meaning no Gurubhai pairs can ever occur under Ustad Samiran.
Fact 3 defines the valid Gurubhai pairs. Since Ananya & Devendra, as well as Bhaskar & Charu, were never Gurubhai, the remaining valid pairs are: Ananya & Bhaskar, Ananya & Charu, Bhaskar & Devendra, and Charu & Devendra. Each of these 4 pairs trained together for exactly 2 years.

Step 3: Assign students to Acharya Raghunath (AR)
Fact 4 states that in 2013, Ananya started under PM and Bhaskar started under US.
Since AR is only available during the 2013-2014 and 2019-2020 slots, and Ananya and Bhaskar are already occupied with other Gurus in 2013, Charu and Devendra must be the ones training under AR from 2013 to 2014. This assignment perfectly accounts for Charu and Devendra being Gurubhai for 2 years.
By process of elimination, Ananya and Bhaskar must take their mandatory turn under AR from 2019 to 2020, which accounts for their 2-year Gurubhai status.

Step 4: Determine Charu's timeline under Pandit Meghnath (PM)
We know Ananya and Charu must be Gurubhai for exactly 2 years. Since they cannot be Gurubhai under Ustad Samiran (he trains individually) or Acharya Raghunath (they took different time slots), this 2-year overlap must occur under Pandit Meghnath.
Ananya started her training under PM in 2013.
If PM had a 3-year span, Ananya would train under PM from 2013 to 2015. Charu is busy with AR until the end of 2014, so the earliest she could start PM is 2015. This would only allow a 1-year overlap (2015), violating the strict condition that they must be Gurubhai for exactly 2 years.
Therefore, Pandit Meghnath must have a 4-year span. Ananya trained under him from 2013 to 2016.
For Charu to overlap with Ananya for exactly 2 years under PM, Charu must train under PM during Ananya's final two years: 2015 and 2016.
Thus, Charu began her 4-year training under Pandit Meghnath in 2015 (which spans 2015 to 2018).

This logically satisfies all conditions, proving that Charu started her training under Pandit Meghnath in 2015.