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# Logic Question-- Classic weighing puzzle

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You have 10 boxes with 1000 coins each, either all real or all fake. Real coins are 16 ounces, fake ones are 17. You have a scale, so you put some number of coins on the scale, and then it tells you the weight of those coins. Your job is to figure out which boxes have fake coins and which ones have real coins in one press of the scale. How do you do this?

Sep 25, 2021

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"You have 10 boxes with 1000 coins each, either all real or all fake. Real coins are 16 ounces, fake ones are 17. You have a scale, so you put some number of coins on the scale, and then it tells you the weight of those coins. Your job is to figure out which boxes have fake coins and which ones have real coins in one press of the scale. How do you do this?"

Those are very heavy coins

Ok I have this ( and I didn't cheat)    It will help you enormously if you can think in base 2 (which I can)

Let the boxes be called box0, box1........ box9

Take 2^(the box number) coins from each box.

2^0=1 from box0

2^1=2 from box1

up to

2^9=512 from box 9

altogether you will have 1023    I know this because  2^10-1=1023  but you can just add them all up or use a GP formula if you want.

Let R be the number of real coins and F be the number of fake coins.   And let the weight be W

We know that   F+R=1023    so    R=1023-F  (1)

and we know   17F+16R= W  (2)

solve these simultaneously and we get   F=W-16368

So now you know exactly how many fakes that you have weighed.  But which boxes do they belong to?

Again, understanding base two helps a lot.

Take   F and divide it by 2, the if there is a remainder it has to be 1 and that would mean that the 0 box is Fake

Take the answer and divide it by 2 again and if there is a remainder then box 1 is Fake

keep going until the whole part is zero then you will have worked out which boxes are fake.  The other boxes are real.

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ex

Say the weight is 16660oz

The number of fake coins is 16660-16368= 292

292 divided by 2 = 146 Remainder 0    so box 0 is not fake

146 divided by 2 =  73 Remainder 0     so box 1 is not fake

73 divided by 2 =  36 Remainder 1     so box 2   IS  fake

36 divided by 2 =  18 Remainder 0     so box 3   is not fake

18 divided by 2 =  9  Remainder 0     so box 4 is not fake

9 divided by 2 =  4 Remainder 1     so box 5   IS  fake

4 divided by 2 =  2 Remainder 0     so box 6 is not fake

2 divided by 2 =  1 Remainder 0     so box 7 is not  fake

1 divided by 2 =  0 Remainder 1     so box 8   IS  fake

So boxes 2, 5 and 8 are full of fake coins and

boxes 0, 1, 3, 4, 6, 7 and 9   are full of real coins

I will leave the checking to you Sep 25, 2021