Eshed, I really appreciate your mathematical efforts, but I'd decipher Egyptian hieroglyphs more easily!

JD has probably made the most sensible contribution here, and he/she's probably right in implying that we're (I'm?) wasting our time chasing a mathematical solution with unavailable starting off figures. We could just be playing these nice little things, instead.

Fortunately for you, I taught some math to 7th graders (12/13 years old) so we'll fix this in a second, but first let me tackle the latter claim.

While JD is absolutely correct, your interpretation is incorrect. It doesn't matter if it's serial numbers of just liliputs or all melodeons, As long as

1) We're only comparing liliput serial numbers

2) Our assumption that construction rates were mostly constant is true

our calculations shouldn't change at all.

Back to the explanation:

In some moment in time, probably 1935, Hohner started giving serial numbers to liliputs.

That's the leftmost point in our axis.

We assume that each Monday the boxes made during the previous week were sent to the dealers.

Let's assume (the very unfounded assumption) that the number of boxes made each week stays mostly the same, and denote it with W.

So the first delivery, on the first Monday after Hohner started giving serial numbers, contained all boxes with serial numbers between 1 and W.

We have info about two Liliputs, the one in the batch of 04/11/35 and the one in the batch of 18/11/35, we don't know however which boxes were in each of these batches other than the specific serials we have.

Let's mark the first serial in the batch of the 4th with N. Since this batch has W boxes, its last serial should be N+W-1. The batch of the 11th should contain boxes from N+W to N+2W-1 and the batch of the 18th should contain boxes from N+2W to N+3W-1. From now on I will ignore the -1s since they're inconsequential.

Now let's see where our two serials, 145913 and 150177, can be in those batches. The first extreme is that 145913 was the

**first** box in the batch of the 4th and 150177 was the

**last** box in the batch of the 18th:

We get that 3W=150177-145913=4264, we divide by 3 and then get W=1421. This is our first bound - since we picked the extreme case, we know that W cannot be smaller than 1421.

Similarly, we can check the other extreme, where 145913 was the

**last** box in the batch of the 4th and 150177 was the

**first** box in the batch of the 18th:

We get that W=4264. This is our second bound, we know that W cannot be larger than 4264.

Bounds are nice and all, but what we actually want is the most probable value of W.

If we assume that our liliputs were picked at random from the batches, on average the first serial is in the middle of its batch: N+W/2 and similarly the second serial should be on average N+2W+W/2 (I know this is hand waving, but there's real math behind this, from the fact that the sum of means is equal to the mean of the sum).

This gives us 2W=4264 and W=2132.

So let us conclude what we have found out:

Every week Hohner produced between 1421 and 4264 boxes. Our best guess regarding the actual number is 2132.

If anything of the above wasn't clear, feel free to ask.