In a two player game, in alternate turns, each player is to pick up 1 to 5 chocolates from a pile of 100. How to win by picking up the last chocolate?

#### The riddle

You are playing a game with your friend Rukmini. Each of you will pick up in alternate turns 1 to 5 chocolates from a pile of 100 chocolates. The player who picks up the last chocolate will win the prize of a chocolate bar that both of you love.

You are to decide who to start the pickup game.

What will be your decision to win the game, and why?

**Time to reach your decision:** 20 minutes.

### Solution to pick up the 100th chocolate riddle

Right at the start, you realize the futility of trying to imagine possible combinations of number of chocolates picked up in alternate turns starting from the beginning. Suppose in the first turn, the first player picks up 1 chocolate piece. Against this pickup number, the second player may pick up 5 different numbers of pieces. In the same way, for 5 different numbers picked up in the first turn, there will be 25 different combinations of pickup number combinations in the second turn. *This cannot be the right way to the solution.*

Instead, you decide to analyze how you can win the game. After all, your main goal is to win. So you assume that you must have won. Now you are on the job of understanding **what it takes to win AT THE END.**

** What must be the last stage situation for you to win?** That's what you try to imagine.

#### Last stage condition for you to win pick up the 100th chocolate riddle game

The last turn must be yours. What about the turn before the last? *How many chocolates should be left so that the other player CANNOT STOP YOU TO WIN at the next turn?* You realize,

Conclusion 1:The number of chocolates in the last but one turn of the other player must be just beyond the reach of the other player.If the number left is between 1 to 5, the other player will grab all the chocolates and the chocolate bar. So the number must be more than 5.

It should at least be 6.

**This is a breakthrough.** You have found the **minimum number of chocolates that must be left by you for the penultimate** (one before the last) **turn** of the other player.

Can you leave more? *What happens if you leave 7 chocolates for the other player?*

He will simply pick up just 1 chocolate and leave 6 chocolates for you. In other words, you fall into the trap you are trying to set up for the other player. With 6 chocolates left, you will pick up 1 to 5 chocolates leaving 5 to 1 chocolates for the other player to grab and win in his next turn.

With this mental trial, you realize,

Conclusion 2:You must leave 6 and only 6 numbers of chocolates for the penultimate turn of the other playerso that he leaves 1 to 5 chocolates for you to pick up on your last turn and win the game.

It follows,

Conclusion 3:You must pick up the 94th chocolate before your last turn. That will leave 6 chocolates for your competitor.

#### Discover the repetitive key pattern that will win you the pick up 100th chocolate game

*How will you make sure to pick up the 94th chocolate?*

This is when **you realize the similarity between making sure to pick up the 100th chocolate and also the 94th.**

To be sure of picking up the 100th chocolate, you must make sure of picking up the 94th chocolate.

**Question:** *To make sure of picking up the 94th chocolate, which chocolate you must pick up before this?*

*The same concept of keeping the 100th chocolate just beyond the reach of your friend now applies to keeping the 94th chocolate just beyond the reach of your friend as well.*

Conclusion 4:To make sure of picking up the 94th chocolate, you must pick up 94 minus 6 or 88th chocolate.

Conclusion 5:You will be able to pick up the 100th chocolateif at every turn you fix your next target chocolate 6 chocolates ahead.

This must be the **key formula that you will apply right from the start for winning the game.**

#### Work backwards from the winning end move to the beginning move

You don't know yet how to start or whether to start at all. But you know for sure how to end and also *discovered the winning formula for deducing the moves before the final move.*

Armed with this knowledge, it is simple to deduce** the number of chocolates that you must pick up from end to the beginning as,**

100th ← 94th ← 88th ← 82nd ← 76th ← 70th ← 64th ← 58th ← 52nd ← 46th ← 40th ← 34th ← 28th ← 22nd ← 16th ← 10th ← 4th.

**You decide to start first and confidently pick up the first 4 chocolates in your first turn.** Thus you will make sure to pick up the 10th, 16th, 22nd....and at the end the 100th chocolate as well to win the game.

It has all been finding the key repetitive pattern for you to win the game.

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