Quiz-1
Queen Ratnamanjuri had a will written that described her fortune of ratnas (precious stones) and also included a puzzle. Her son Khoisnam and their 99 relatives were invited to the reading of her will.
She wanted to leave all of her ratnas to her son, but she knew that if she did so, all their relatives would pester Khoisnam forever. She hoped that she had taught him everything he needed to know about solving puzzles.
She left the following note in her will—
“I have created a puzzle. If all 100 of you answer it at the same time, you will share the ratnas equally. However, if you are the first one to solve the problem, you will get to keep the entire inheritance to yourself.
Good luck.” The minister took Khoisnam and his 99 relatives to a secret room in the mansion containing 100 lockers. The minister explained—
“Each person is assigned a number from 1 to 100.
• Person 1 opens every locker.
• Person 2 toggles every 2nd locker (i.e., closes it if it is open, opens it if it is closed).
• Person 3 toggles every 3rd locker (3rd, 6th, 9th, … and so on).
• Person 4 toggles every 4th locker (4th, 8th, 12th, … and so on). This continues until all 100 get their turn.
In the end, only some lockers remain open. The open lockers reveal the code to the fortune in the safe.”
Before the process begins, Khoisnam realises that he already knows which lockers will be open at the end. How did he figure out the answer?
Write the locker numbers that remain open.
“The passcode consists of the first five locker numbers that were touched exactly twice.”
Which are these five lockers?
Hint: Find out how many times each locker is toggled.
SOLUTION-1
🏰 कहानी और Puzzle का सार
100 lockers , numbered 1 से 100 तक।
शुरू म सारे lockers बंद ह।
100 लोग (Khoisnam + 99 relatives), हर एक को 1 से 100 तक का number
दया गया है।
Rule: Person n हर n-th locker को toggle (खुले को बंद, बंद को खुले) करता है।
Person 1 → हर locker खोलेगा (क्यक सब शुरू म बंद ह)
Person 2 → हर 2nd locker toggle करेगा (2, 4, 6, …)
Person 3 → हर 3rd locker toggle करेगा (3, 6, 9, …)
और इसी तरह Person 100 → सफ 100th locker toggle करेगा।
Hint: Find out how many times each locker is toggled.
🔍 Logic Behind the Puzzle
हर locker को उसक े factors (divisors) क े हसाब से toggle कया जाता है।
Example: Locker 12 → factors: 1, 2, 3, 4, 6, 12 → total 6 times toggle
होगा।
अगर toggle even बार होगा → locker बंद रहेगा।
अगर toggle odd बार होगा → locker खुला रहेगा।
Odd number of factors सफ perfect squares क े होते ह (क्यक एक factor
repeat होता है, जैसे 4 × 4 = 16)।
✨ Final Result
So at the last perfect square numbered lockers खुले हगे:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
💡 Safe Code
Queen ने code को इन्ह open lockers म छ ु पाया था।
Open lockers क े numbers = [1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
अगर puzzle कहता है “सफ open lockers पढ़ो” → यही code है।
SOLUTION-1A
Khoisnam immediately collects word clues from these 10 lockers and
reads,“The passcode consists of the first five locker numbers that were touched exactly twice.”
Which are these five lockers?
The lockers that are toggled twice are the prime numbers, since each prime number has 1 and the number itself as factors. So, the code is 2-3-5-7-11.
Step 1 — Toggle rule / नयम हर person numbered k हर k‑th locker को toggle करता है (याने open ↔
close)।
(English: Person numbered k toggles every k‑th locker.)
Step 2 — कब एक खास locker toggle होगा? कसी दए हुए locker नंबर nn को तब‑तब toggle मलेगा जब कोई person kk ऐसा हो क kk उस locker का divisor हो (यान k divides n). उदाहरण: person 3 locker 12 को toggle करेगा क्यक 3 divides 12.
(English: Locker nn is toggled on each pass number kk that is a divisor of n.)
Step 3 — इसलए total toggles = number of divisors
इसलए कसी locker nn पर क ु ल जतने बार touch (toggle) होगा, वही nn क े positive divisors क संख्या होगी। इसे हम τ(n)τ(n) या “number of divisors” कह सकते ह।
(English: So the total number of toggles on locker nn equals the number of positive divisors of nn.)
Step 4 —
“exactly twice touched” का मतलब यद कोई locker exactly 2 बार touch हुआ है, तो उसक े क े वल दो ही divisors होने चाहए। क े वल दो divisors वाले numbers वही होते ह जनको हम prime numbers कहते ह (दो divisors = 1 और खुद‑का number)।
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