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This is follow up question of 99 chips into more chips, which is originally:

You are at a casino in Vegas and you have earned 99 chips by playing poker!

While you are checking out a slot machine, someone comes to you and congratulates you that you have a chance to make more chips by using your own chips into a strange four slot machine.

With this machine, you can put as many coins as you want into the four slots available and pull the trigger only once to make more coins but all four slots where you put your coins in the machine behave differently:

  • One of them makes your coins four times as many as before!
  • Another slot just gives your coins back.
  • The last two slots do not give your coins back at all.

But you do not know which slot is which and you can take your coins back after pulling the trigger from somewhere else as a whole.

All parts are the same (you still have 99 chips etc) but instead of "pull the trigger only once" rule, you have a chance to pull the trigger twice. With the new given opportunity,

At most how many coins can you guarantee to have at the end when playing with this machine?

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  • $\begingroup$ The original definition of the machine does no longer fit the new behavior: if you pull the trigger a second time, the second slot does not 'give your coins back', because those coins have been given back with the first pull! And you do not allow to put coins into slots for the second time, but just pull the trigger twice. So on the second pull the slot returns another bunch with the same number of coins. $\endgroup$
    – CiaPan
    Commented Feb 5, 2018 at 12:09

2 Answers 2

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The maximum that can be guaranteed is

468 chips

Using this method to find the 4x slot and also maximize winning on the first round.
Insert a number of chips into each slot: 23, 24, 25, 26 and save one.

According to how much we have after one pull, identify the 4x slot.

slot | | winning
4100 116
4010 117
4001 118
1400 119
0410 121
0401 122
1040 123
0140 124
0041 126
1004 127
0104 128
0014 129
Then we are guaranteed to have 116 + 1 chips at least, so $117 \times 4 = 468$ for the second pull.

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  • $\begingroup$ good answer thanks, I believe this question is a good example that with one word change, all methodology and the result can change everything in the question :) $\endgroup$
    – Oray
    Commented Feb 4, 2018 at 16:19
  • $\begingroup$ But the problem statement does not include putting coins into slots for the second time! It only allows to pull the trigger twice. $\endgroup$
    – CiaPan
    Commented Feb 5, 2018 at 12:11
  • $\begingroup$ @CiaPan If that was the case then the answer would be the same as for one pull, but do it twice. $\endgroup$
    – Jay
    Commented Feb 5, 2018 at 13:08
  • $\begingroup$ Sure! But, strictly speaking, that's how the problem was stated. In case of a vague question you can make an answer more precise by adding a note about specific interpretation of a problem statement. (I apologize, if my comments may sound a bit rude. I intend to express my opinion rather than a recommendation, but my English isn't flexible enough, yet.) $\endgroup$
    – CiaPan
    Commented Feb 5, 2018 at 13:22
  • $\begingroup$ @CiaPan pull the trigger twice does not necessarily mean you need to pull twice in a row. you can collect chips then pull it again. that will make it twice too. $\endgroup$
    – Oray
    Commented Feb 6, 2018 at 6:50
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The maximum should be

four times the answer of the original question: $4\times 123=492$. First pull: 24,24,24,24 (keep 3) Second pull: all 123 in the 4x slot

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  • $\begingroup$ u dont know which slot which without counting chips. $\endgroup$
    – Oray
    Commented Feb 5, 2018 at 5:25

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