# Solution

My friend should get the following four weights: 1 kg, 3 kg, 9 kg and 27 kg. This will allow him to measure all whole kilograms up to 40 kg.

This is how he will go about it:

1 kg: He already has 1 kg.

2 kg: He places 3 kg on one side of the balance and 1 kg on the other side. Thus, 2 kg = 3 kg - 1 kg.

3 kg: He already has 3 kg.

4 kg: 3 kg + 1 kg.

5 kg: 9 kg - (3 kg + 1 kg). So, he places 9 kg on one side of the balance, and 3 kg and 1 kg weights on the other side.

6 kg: 9 kg - 3 kg.

7 kg: (9 kg + 1 kg) - 3 kg.

8 kg: 9 kg - 1 kg

9 kg: He already has 9 kg.

10 kg: 9 kg + 1 kg.

11 kg: (9 kg + 3 kg) - 1 kg.

12 kg: 9 kg + 3 kg.

13 kg: 9 kg + 3 kg + 1 kg.

14 kg: 27 kg - (9 kg + 3 kg + 1 kg).

15 kg: 27 kg - (9 kg + 3 kg).

16 kg: (27 kg + 1 kg) - (9 kg + 3 kg).

17 kg: 27 kg - (9 kg + 1 kg).

18 kg: 27 kg - 9 kg.

19 kg: (27 kg + 1 kg) - 9 kg.

20 kg: (27 kg + 3 kg) - (9 kg + 1 kg).

21 kg: (27 kg + 3 kg) - 9 kg.

22 kg: (27 kg + 3 kg + 1 kg) - 9 kg.

23 kg: 27 kg - (3 kg + 1 kg).

24 kg: 27 kg - 3 kg.

25 kg: (27 kg + 1 kg) - 3 kg.

26 kg: 27 kg - 1 kg.

27 kg: He already has 27 kg.

28 kg: 27 kg + 1 kg.

29 kg: (27 kg + 3 kg) - 1 kg.

30 kg: 27 kg + 3 kg.

31 kg: 27 kg + 3 kg + 1 kg.

32 kg: (27 kg + 9 kg) - (3 kg + 1 kg).

33 kg: (27 kg + 9 kg) - 3 kg.

34 kg: (27 kg + 9 kg + 1 kg) - 3 kg.

35 kg: (27 kg + 9 kg) - 1 kg.

36 kg: 27 kg + 9 kg.

37 kg: 27 kg + 9 kg + 1 kg.

38 kg: (27 kg + 9 kg + 3 kg) - 1 kg.

39 kg: 27 kg + 9 kg + 3 kg.

40 kg: 27 kg + 9 kg + 3 kg + 1 kg.

• 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