# Fill your bags with marbles

You have 50 bags. How many marbles do you need at least, so that you can have a different number of marbles in each bag?

• Sorry but Im voting to close this as this is more of a textbook-style question than a puzzle. Happy puzzling tho! – Omega Krypton May 3 at 1:28
• @OmegaKrypton Vg fbhaqf yvxr n chmmyr gb zr vs lbh vagrecerg vg jvgu n yngreny-guvaxvat engure guna n zngurzngvpf gnt. Znlor gur BC pna pynevsl. – Anon May 3 at 1:33
• sorry, retracting close vote after seeing answer! nice one! – Omega Krypton May 3 at 1:42
• [lateral thinking]Surely this depends upon the number of marbles which were in each bag to begin with[/lateral thinking] – Trevor Powell May 3 at 7:53

The minimum number of marbles is:

$$49$$. Put a single marble in $$49$$ of the bags, and then put all the bags one inside each other, and then put the empty bag inside them all. In other words, going from outside to in, we will have bag-marble-bag-marble-...-bag-marble-bag, and the sequence of the number of marbles in each bag going from inside to out will be $$0,1,2,3,...,49$$.

We can prove this is the minimum number because:

For there to be a different integer number in each bag, the sequence that gives the smallest possible number in the bag with the most marbles is $$0,1,2,...,49$$, so at least one bag must have $$49$$ marbles in it.

If we increase the laterality of our thinking even further (with weakened respect for dimensional constraints, and tongue lightly pressed in cheek), we can improve our solution to:

$$48$$ marbles. Set up the first $$49$$ bags as above with the $$48$$ marbles, so that they contain $$0,1,2,...,48$$ marbles each. Then alter the final bag so that it has the topology of a Klein bottle, and set it down where you please; it now contains every marble in the universe, since its interior is indistinguishable from its exterior. Our bags thus contain $$0,1,2,...,47,48,|M|$$ marbles where $$M$$ is the set of all objects in the universe that could properly be construed as marbles, whose size (given that I also happen to own a marble, and you have $$48$$) is strictly greater than $$48$$.

Additionally, with tongue now firmly planted in cheek, the minimal number can be drastically reduced to:

$$1$$ with a simple repeated application of the Banach-Tarski theorem.

• That was quick : ) – Eric May 3 at 1:35
• @Eric rot13(Unun, srry serr gb npprcg gur nafjre vs vg jnf jung lbh jrer trggvat ng, be jnvg gb frr vs nalbar ryfr pbzrf hc jvgu n orggre yngreny guvaxvat fbyhgvba). – Anon May 3 at 1:37
• What happened to your comments? They seem to be garbled and unrecognizable? – Eric May 3 at 1:40
• @Eric You can use rot13 for spoilers in the comments. – Anon May 3 at 1:43
• YBY Guvf Onanpu-Gnefxv ersrerapr znqr zl qnl! – Galen May 3 at 2:08