June 2024
Problem of the Month

Five Numbers
Submitted by K. Sengupta, Calcutta, India

With five numbers there are 5 ways to group 4 of them together.   If you take the averages of each group of 4 numbers you get...

3428.5
3425.75
3398
3120.25
342.5

What are the 5 numbers?

Solution:

The numbers are 1, 12, 123, 1234, and 12345

Let the five numbers be A, B, C, D and E, with A < B < C < D < E.

Clearly, A+B+C+D must then be the lowest group sum followed by A+B+C+E,
while B+C+D+E must be the highest group sum, preceded by A+C+D+E.
That means the middle number must be A+B+D+E.

Matching these up in the problem, we obtain five equations:
(A+B+C+D)/4 = 342.5 .... (i)
(A+B+C+E)/4 = 3120.25 .... (ii)
(A+B+D+E)/4 = 3398 ...... (iii)
(A+C+D+E)/4 = 3425.75 .... (iv)
(B+C+D+E)/4 = 3428.5 ..... (v)

Adding (i) thru (v), we obtain:
A+B+C+D+E = 13715 (vi)

Multiplying (i) thru (v) by 4 in turn and separately subtracting these from (vi), we obtain:

E = 12345
D = 1234
C = 123
B = 12
A = 1

Thus, the required numbers (in ascending order) are 1, 12, 123, 1234 and 12345

Correctly solved by:

1. Kamal Lohia	Holy Angel School, Hisar, Haryana, India
2. Dr. Hari Kishan	D.N. College, Meerut, Uttar Pradesh, India
3. Seth Cohen	Concord, New Hampshire, USA
4. Davit Banana	Istanbul, Turkey
5. Eloise Michael	Smith Academy, Hatfield, Massachussetts, USA
6. Pam Slone	Thacker, West Virginia, USA
7. Kelly Stubblefield	Mobile, Alabama, USA

Send any comments or questions to: David Pleacher