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# Insert Coin Here

## Question

Your light and heavy cheat coins weigh like 2 proper coins
but looks the same.They always hit the jackpot in any
slot machines. Drowsy, you accidentally used them on
a vending machine along with 2 other proper coins.
However the coins malfunctioned the machine causing it
to dispense hot cup of coffee with the inserted coin inside.
To test weights, you may insert a coin using the beam-
balance machine once.

\_/ \_/ \_/ \_/

___________ __________
|_________________|
____/\____
/ -- \
[----------------] o o

A) With 2 spare coins in pocket,how can you identify the cheat coins ?
B) How many spare coins is needed if the first trial is balanced ?

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Let's number the 4 suspect coins 1, 2, 3 and 4 and let's use the notation H-heavy, L-light and N-normal. Among the 4 suspect coins we have 12 possible distributions: HLNN, LHNN, NLHN, NHLN, NNLH, NNHL, HNNL, LNNH, HNLN, LNHN, NLNH, NHNL.

We also have 2 known normal coins (NN). Let's use one of the known normal coins to pay for the first use of the scale and weigh 1,2 vs. 3,N. We will use the other known normal coin in the first weighing. The possible distribution of outcomes from the first weighing is

Balanced: HLNN, LHNN, NLHN, NHLN

Left is heavy: HNLN, HNNL, NNLH, NHNL

Right is heavy: LNHN, LNNH, NNHL, NLNH

If balanced, we know that the coin 4 is normal and the cheat coins are either 1,2 or 2,3. We can now use the second known normal coin for the second and final weighing. We will weigh 1,2 vs.3,4. If balanced then 1,2 is the cheat pair. If not, then 2,3 is the cheat pair. Done in 2 weighings.

If left or right is heavy we will need 1 or 2 more weighings to identify cheat coins, but we need to identify one normal coin in the second weighing to use for the third. Let's look at the case when the left side was heavy. It's symmetrical, so the same logic can be applied for the right side too.

Let's make the second weighing 1,4 vs 2,3.

Balanced: HNNL. 1 and 4 are the cheat coins. Done in 2 weighings.

Left: HNLN or NNLH. Need the third weighing, but we now know that 2 is a normal coin. Use it to weigh 1 vs 4 and identify the cheat pair. Done in 3 weighings.

Right: NHNL. 2 and 4 are the cheat coins. Done in 2 weighings.

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I assume the vending machine returned all 4 deposited coins; and the balance takes one coin for each use.

Drop returned coins onto the balance beam, two on each side and pay for the weighing with one of the coin out of your pocket. If balanced, then one of the pairs is the cheating pair. Waste no more money on weighting and go to the slot machines. Pick the one with the biggest payoff and test the two pairs of coins there.

If one side was heavier than the other, take one coin from each pan and swap them. Weigh again using the last coin from your pocket. If the balance reverses (the side that was heavier becomes lighter) - then the swapped coins are the cheating pair. If the side that was heavy stays heavy - the swapped coins are regular. If the two sides balance - go to the slot machines.

Good casinos give you complementary drinks when you gamble. There was no need to use vending machine for coffee.

Edited by Prime
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That's right Prime they were all returned inside the coffee cups.
The trouble with testing the coins in slot machine is ..the proper coins can hit jackpots too.
Also the arcade owner is low on profit so every machine shall be coin operated.

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1. Acquire a waterproof marker (or in this case, a coffeeproof marker).

2. Mark the two spare coins from your pocket with the marker.

4. Insert two of the unmarked coins into the vending machine along with both marked coins.

a. If the machine malfunctions, continue inserting combinations of unmarked coins with the two marked coins while enjoying your free coffee.

b. If the machine fails to malfunction, the two remaining coins are your cheat coins.

i. Go to Vegas.

ii. Win.

Edited by BobbyGo
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1. Acquire a waterproof marker (or in this case, a coffeeproof marker).

2. Mark the two spare coins from your pocket with the marker.

4. Insert two of the unmarked coins into the vending machine along with both marked coins.

a. If the machine malfunctions, continue inserting combinations of unmarked coins with the two marked coins while enjoying your free coffee.

b. If the machine fails to malfunction, the two remaining coins are your cheat coins.

i. Go to Vegas.

ii. Win.

1. Yes you can acquire the marker! But the marker/pen vendo say " Insert 2 Coins Here"

4.a. The Coffee machine is already malfunctioned ..people are in queue for the free

b. If the machine fails to malfunction(because of the cheat coins), the cheat coins may not be dispensed anymore.

i. Win more coins first on the arcade, need some transport money.

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That's right Prime they were all returned inside the coffee cups.

The trouble with testing the coins in slot machine is ..the proper coins can hit jackpots too.

Also the arcade owner is low on profit so every machine shall be coin operated.

I wouldn’t classify winning with regular coins as “problem”. Just take the winnings home and measure the remaining two coins at leisure using your own precision scale.

But if you insist on using the coin operated balance to the end...

Use one out of pocket coin to pay and weigh the other out of pocket coin against one of the returned coins.

1) If they weigh the same, you’ve got two known regular coins. Pay with one of them and weigh another against one of the remaining 3 returned by the vending machine. If same, the remaining 2 returned coins are your cheating pair. If not - you have found one counterfeit and know which one (heavier or lighter). Set the found counterfeit coin aside pay with your known good coin to weigh the remaining 2 returned coins against one another thus finding the second fake coin. And you still have one spare regular coin, which could be used in another vending machine.

2) If the first trial reveals a counterfeit and its fault (heavy or light), it’s going to cost even less. Use your good coin to pay and weigh 2 of the unknown 3 coins against one another. If they weigh the same, they are all good, so the third coin is the remaining fake. If different, since you know the remaining bogus coin’s fault (lighter or heavier), you can pick it off the balance. Now you have your trick pair, two true coins, and a cup of coffee.

Note, the best case scenario here costs two standard coins for weighing and the worst - three. With my previous method the worst case scenario loses all four standard coins, two for the weighing and two in gambling. The best case costs just one weighing.

All that does not matter, considering you win jackpot, and you have resigned to spend four coins in the coffee machine anyway.

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Simplifying 3 weighings solution.

: if tilted
13 12 > 12 11
: try heavier to identify heavy & proper coin
13 > 12
: use known proper coin to identify the lighter coin
11 < 12

:if balanced
11 13 = 12 12
: try any side
12 = 12
: use known proper coin to identify the heavy and light coin
13 > 11

=====================

Both approached of K-man and Prime are CORRECT !
Nevertheless they consumed 3 coins on weighing.

======================
They're the key to cheaper way.
using coin for weighing is
just an option..for checking

Here is a how \_/ \_/ \_/ \_/

fill up all other cups from cup 2.
drink the remaining to stay awake.
now empty with coin inside..
put cup 2 in cup 1, let overflow.
take cup 2 out and put it down.
we have now coffee level for coin 2,

we know that the weight of volume
of coffee displaced is equal to the
weight of sumerged volume of cup.
-bouyancy principle.

next, empty cup 3 into cup 2
put cup 3 in cup 2, let overflow.
take cup 3 out and put it down.
we have now coffee level for coin 3,

next, empty cup 4 into cup 3
put cup 4 in cup 3, let overflow.
take cup 4 out and put it down.
we have now coffee level for coin 4,

this somehow revealed the relative

weights of 4 coins. the higher the lighter.
drink the remaning coffee to keep aware

and check the any coins using the

balance machine once.

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Here is a how \_/ \_/ \_/ \_/

fill up all other cups from cup 2.

drink the remaining to stay awake.

now empty with coin inside..

put cup 2 in cup 1, let overflow.

take cup 2 out and put it down.

we have now coffee level for coin 2,

we know that the weight of volume

of coffee displaced is equal to the

weight of sumerged volume of cup.

-bouyancy principle.

next, empty cup 3 into cup 2

put cup 3 in cup 2, let overflow.

take cup 3 out and put it down.

we have now coffee level for coin 3,

next, empty cup 4 into cup 3

put cup 4 in cup 3, let overflow.

take cup 4 out and put it down.

we have now coffee level for coin 4,

this somehow revealed the relative

weights of 4 coins. the higher the lighter.

drink the remaning coffee to keep aware

and check the any coins using the

balance machine once.

Nice solution using the Archimedes' principle, TSLF. However, from the OP it wasn't clear that you actually had 4 cups of coffee to work with. I thought you only had one cup, which would preclude such a solution. Nice puzzle nonetheless.

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