Holmes wrote:Oh My God! Such a Long time after the last Lateral Thinking Problem, I had to find a new one, maybe the old Masters of Reasoning will come out!

Here it is:

** You have twelve marbles. Eleven of the marbles are of equal weight, but one is heavier or lighter. You have a balancing scale you can use to find this marble and figure out if it weighs more or less than the others. What is the minimum number of weighings required to do this?**

My answer is:

[spoiler]An average of 3.083 weighings[/spoiler]

because:

[spoiler]

Okay, this is the best I could do to describe the procedure:

Step 1) Take marbles [1,2,3,4] and weigh them against marbles [5,6,7,8]. Leave marbles [9,10,11,12] off the balance

If it balances, then you know the odd marble is in the set [9,10,11,12]. Go to Step 2a.

If it doesn't balance, record whether [1,2,3,4] is heavier or lighter than [5,6,7,8]. Go to Step 2b.

Step 2a) Weigh marbles [9,10] against marbles [11,1]. Leave marble [12] off the balance

If it balances, you know the odd marble is [12]. Got to Step 3a.

If it doesn't balance, record whether [9,10] is heavier or lighter than [11,1], Go to step 3b.

Step 3a) Weigh marble [12] against marble[1]. If [12] is heavier, than the odd marble is heavier. Otherwise it's lighter.

Done! (3 weighings 8.3% occurance)

Step 3b) Weigh marble [9] against marble [10]. Leave marble [11] off the balance.

If it balances, you know that marble [11] is the odd marble, and that it's heavier/lighter as observed in step 2a.

Done! (3 weighings 8.3% occurance)

If it doesn't balance, then if [9,10] in step 2a was heavier, the heavier marble is the odd marble.

Otherwise the lighter marble is the odd marble.

Done! (3 weighings 16.7% occurance)

Step 2b) Weigh marbles [1,2,3] against marbles [4,11,12].

If they balance, then the odd marble is in the set [5,6,7,8]. Go to step 3c.

If they don't balance and [4,11,12] matches being heavier/lighter as [1,2,3,4] was to [5,6,7,8] in step 1

then marble [4] is the odd marble and is heavier/lighter as observed.

Done! (2 weighings 8.3% occurance)

If they don't balance and [1,2,3] matches being heavier/lighter as [1,2,3,4] was to [5,6,7,8] in step 1

then Go to Step 3d.

Step 3c) Weigh marbles [5,6] against marbles [7,1]. Leave marble [8] off the blanace

If they balance, then the odd marble is [8] and it is heavier/lighter as observed in step 1.

Done! (3 weighings 8.3% occurance)

If they don't balance and [7,1] matches being heavier/lighter as [5,6,7,8] was to [1,2,3,4] in step 1,

then [7] is the odd marble, and is heavier/lighter as observed in.

Done! (3 weighings 8.3% occurance)

If they don't balance and [9,10] matches being heavier/lighter as [5,6,7,8] was to [1,2,3,4] in step 1,

then go to step 4.

Step 4) Weigh marble [9] against marble [10].

The marble that matches being heavier/lighter as [5,6,7,8] was to [1,2,3,4] in step 1 is the odd marble,

and is heavier/lighter as observed.

Done! (4 weighings 16.7% occurance)

Step 3d) Weigh marble [1] against marble [2]. Leave marble [3] off the balance.

If they balance, then marble [3] is the odd marble and is heavier/lighter as observed in step 1.

Done! (3 weighings 8.3%)

If they don't balance and if [1,2,3,4] was heavier in step 1, then the heavier of [1] and [2] is the odd marble.

Otherwise the lighter of [1] and [2] is the odd marble.

Done! (3 weighings 16.7%)

Average weighings: 3.083

Unfortunately, I can't find a way to get it down to an average of 3, or to avoid a fourth weighing somewhere.

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