 ERJ Brainteaser: Answers supplied for Lighter batch question

Factory technician Ian has 10 batches of tires, all of which, but one, have tires weighing 100kg. One batch has tires weighing 90kg. Ian is allowed to use the scales only once. How can he spot the odd batch out?

To avoid disputes, we created this special page to present all answers to this weighty teaser:

Take one tire from batch 1, two tires from batch 2, three tires from batch 3 etc. With 100 kg tires in all batches you get a weight of 5500 kg; now weighing the sample (as above described), then deduct this result from 5500 and divide by 10, the you get the batch with the 902 kg tires. Example: Batch 3 = 90 kg, the weight is 5470 kg  5500-5470 = 30  30/10 = 3 (= Batch 3)

OR

Leave batch 10 out. Then the same as below for batch 1-9, gives 45 tires to weight. When all batches have 100 kg tires, then you get a total of 4500 kg; if there is any difference to 4500 kg after weighing all 45 tires, then deduct the result from 4500 kg, divide by 10 and we get the number of the batch. If not, it is batch 10 with the 90 kg tires.

Hans-Bernd Lüchtefeld, market research & communication manager, PHP Fibers GmbH, Germany

He places 1 tire from batch 1 along with 2 tires from batch 2, 3 tires from batch 3, 4 tires from batch 4, 5 tires from batch 5, 6 tires from batch 6, 7 tires from batch 7, 8 tires from batch 8, 9 tires from batch 9 and 10 tires from batch 10.

If the total weight is 5,490kg, then batch 1 is the odd one

If the total weight ends in 5,480Kg, then batch 2 is the odd one

If the total weight ends in 5,470Kg then batch 3 is the odd one

If the total weight ends in 5,460Kg then batch 4 is the odd one

If the total weight ends in 5,450Kg then batch 5 is the odd one

If the total weight ends in 5,440Kg then batch 6 is the odd one

If the total weight ends in 5,430Kg then batch 7 is the odd one

If the total weight ends in 5,420Kg then batch 8 is the odd one

If the total weight ends in 5,410Kg then batch 9 is the odd one

If the total weight is a multiple of 5,400Kg then batch 10 is the odd one

John Droogan, advanced polymers and composites, MegaChem (UK) Ltd, Caldicot, Monmouthshire, UK.

I am not sure If “using” the scale once includes reading the result several times but I would pile all 10 batches on top of each other and then remove one by one. When the number can be divided by 100 without digit after the comma, the previous batch had 90 kg. Ralf Ortmayr, senior pricing analyst, Toyo Tire Europe GmbH, Willich, Germany

Ian should number the batches 1,2,3 and so on up to 10. He should then take 1 tyre from batch 1, 2 from batch 2 and so on up to 10 from batch 10 and weigh the combined total. If all weighed 100kg the total would be 100*[1+2…] =5500kg. The amount by which the total is less than this will identify the batch with the lighter tyres.

John Bowen, consultant, Bromsgrove, UK

You only need to weigh once if you take one tyre from batch 1, 2 tyres from batch 2, through to 9 tyres from batch 9, an no tyres from batch 10. If batches 1-9 contain samples from the 90 kg tyres, then the batch with 90 kg tyres will be identified by the multiple of 10 kg that the total weight is short of 4500 kg, i.e. if 40 kg short the 90 kg tyres will be batch 4. If the total of all samples is 4500 kg, then batch 10 contains the 90 kg tyres.

Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands;

Ian can take one tyre from the first batch, 2 tyres from the second batch , .., 10 tyres from the 10th batch and weight the total. If all the tyres weighed 100 kg the total weight would be 10*(10+1)/2 * 100kg = 5500 kg. If the weight is 10kg less, the light batch is #1, if 20 kg is #2, …, if 100kg is #10

Michele Girardi, Scame Mastaf Spa, Suisio, Italy

Take 1 tire from batch 1, 2 tires from batch 2, 3 tires from batch 3, continuing until you have 10 tires from batch 10. Weigh all of them at once. Subtract the weight from 550 and the result will tell you the batch with the 90 kg tires.

Paul Knutson, textile engineer, Timken Belts, Springfield, Missouri, USA

Take one tyre from the first batch of tyres, two from the second, three from the third etc. then weigh the sum total of these samples. If the batches all weighed 100kg and there were 10 tyres in each batch weighing 10kg each we would expect the total weight to be 550kg. This would scale for any number of tyres so long as there are at least 10 (the number of batches) and that each batch has the same in total. If the sample weighs 549kg then we know the 90kg batch was the first, if it weighed 548kg then it was the second, 547kg the third, and so on.

Andrew Beasley, product analyst, Hankook Tyre UK Ltd, UK

The trick is to take 1 tire of batch 1, 2 tires of batch 2, 3 of batch 3, 4 of batch 4 and so on to 10 tires of batch 10. If all tires were to weight 100 kg the total would be 5500 kg (55 tires total). But if batch 1 has the tires weighing 90 kg then the total will be 5490 kg (because 1 tire weight 90 kg and 54 weight 100 kg each). If batch 2 has the tires weighing 90 kg then the total will be 5480 (2 tires weighing 90 kg each and 53 weighing 100 kg each). So depending on the total weight the batch with the tires weighing 90 kg can be identified with the table below.

Total weight  Batch with 90 kg tires

(kg)    (batch #)

5490   1

5480   2

5470   3

5460   4

5450   5

5440   6

5430   7

5420   8

5410   9

5400   10

Thierry Montcalm, R&D and innovation manager, Soucy Techno, Canada

Put all 10 batches on the scales and then remove one batch at a time. When the weight drop is 90kg then that is the odd batch.

Stephen Fulton, R&D technology development manager, Umicore Specialty Materials Brugge NV, Brugge, Belgium;

It is necessary to take 1 bus from the first batch and mark with number 1. Then take two tires from the second batch and mark with number 2. Then three tires from batch 3 and mark with number 3 etc. Then we weigh all the selected tires. If all the tires weighed 100 kg, then the total weight would be 5500 kg. Therefore from 5500 we will take away weight which has turned out at weighing. And divide this number by 10. As a result, we get the batch number, in which the tires weigh 90 kg each.

Vilen Kancerov, export department director, SIA Latakko, Riga, Latvia