# Weighty problem

One of my friends is a grocer. My friend, to tell you the truth, is somewhat miserly. In this digital age, he continues to use a vintage weighing scale - the kind of scale for which you require those ugly and terrible-looking cast iron weights!

The friend, however, needs another set of weights. He wants to buy only four weights which will allow him to measure from one kilogram upto 40 kilograms in a single weighing. Let me explain what I mean: Suppose the friend buys weights of one kilogram, two kilograms, three kilograms and four kilograms. With these four weights he can measure 1 kg, 2 kg, 3 kg, 4 kg, 5 kg (4+1, or 3+2), 6 kg (2+4 or 1+2+3), 7 kg (3+4, or 1+2+4), 8 kg (1+3+4, or 2+3+4-1), 9 kg (2+3+4), and 10 kg (1+2+3+4). Beyond that he will have to weigh two or more times for larger quantities. For instance, to measure 11 kg, he will first have to weigh out 10 kg and then another 1 kg (or 2 kg and then 9 kg, or any such combination). My friend doesn't wish to weigh multiple times; he should be able to weigh anything from 1 kg to 40 kg in a single weighing. Oh yes, I must not forget to elaborate on the weights combination I have used to measure 8 kg: one of the combinations I have used is 2+3+4-1. This is perfectly alright; I can place the 2 kg, 3 kg, and 4 kg weights on one pan of the scale (that's 9 kg) and place the 1 kg weight on the other pan; this will allow me to weigh 8 kgs.

So, 1 kg, 2 kg, 3 kg and 4 kg weights don't allow my friend to measure from 1 kg to 40 kgs in single weighing. That was just an example I gave to explain my friend's requirement.

The question is: what are the four weights he should acquire which will allow him to measure from 1 kg to 40 kgs (only whole numbers; we don't want to measure quantities like 1.5 kilograms)?

• Union Public Service Commission - www.upsc.gov.in
• IIT-Kharagpur - www.iitkgp.ac.in
• Indian Statistical Institute - www.isical.ac.in
• Indian Institute of Technology Madras - www.iitm.ac.in
• Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
• Indian Institute of Mass Commission - www.iimc.nic.in
• IIT Bombay - www.iitb.ac.in