# ERJ Brainteaser: April

12 Apr 2024

Each month, ERJ sets a weekly brainteaser, with questions of varying degrees of difficulty. Readers supplying the most accurate (and stylish) answers are then considered for the prestigious **Brainiac of the Month** title.

Question 2: **Odd number**

What is the sum of the first 77 odd numbers?

**Answer**: Well done to the following select group who so neatly (Solutions below) got to the correct answer **5929**: **Sudi Sudarshan**, principal consultant, Global Mobility Strategies, USA; **John Bowen**, rubber industry consultant, Bromsgrove, Worcs, UK; **Andrew Knox**, Rubbond International, The Netherlands; **Amparo Botella**, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; **Peter D. Talbot**, research scientist, Chem-Trend LP, Howell, Michigan, USA; and everyone else who had a go.

SOLUTIONS

*Sudi Sudarshan *

*My answer: 5929*

*77th odd number is 77*2-1=153*

*Sum = 1 + 3 + 5 + ... + 153 *

*= (1+153)+(3+151)+ ...+(75+79)+77*

*=154*38+77 = 5852+77 = 5929*

*John Bowen*

*It can be proved that the sum of a series of n odd numbers starting at 1, = n-squared, so the sum of the first 77, starting with 1 , = 77 x 77 = 5929*

*Andrew Knox*

*Answer: 77 squared = 5929*

*Trying the first few sequences and a simple pattern emerges (without recourse to the Internet).*

*1 (3) 5 = 9*

*1 3 ( ) 5 7 = 16*

*1 3 (5) 7 9 = 25*

*1 3 5 ( ) 7 9 11 = 36*

*1 3 5 (7) 9 11 13 = 49*

*Sum appears to be the square of the middle number (or the "interpolated" middle number if there are an even number of numbers in the sequence).*

*The last number appears to be 2n -1 so in a sequence of the 77 uneven numbers this will be 153, and the middle number if n is uneven will be n, i.e. in this case 77.*

*So the sum in this case is the square of 77 = 5929.*

*Amparo Botella*

*The last odd number of the 77 first ones is 153 (1,3, 5, 7……… 149, 151, 153) and the sum of all of them is: 5929*

*Peter D. Talbot*

*First, determine what is the 77th odd number.*

*77th odd number = (77 X 2) – 1 = 154 – 1 = 153*

*Attempt to make an even number of pairs using the highest number added to the lowest number in the series.*

*77 odd numbers will not give a whole number of pairs (38.5).*

*Therefore, make 38 pairs of odd numbers (1st odd number + 76th odd number, 2nd odd number + 75th odd number etc.) and add the 77th odd number to make the total.*

*76th odd number is 151.*

*Each pair of odd numbers total 152 (1 + 151, 3 + 149, 5 + 147 etc.)*

*Total of 77 odd numbers = (38 X 152) + 153 = 5776 + 153 = 5929*

New teaser on Monday.

Question 1: **Give me five (decimal places)**

.1

.04

.001

.0005

.0000?

**Answer**: This time it’s hats off to **Sudi Sudarshan**, principal consultant, Global Mobility Strategies, USA, who came up with the correct answer (see below) before any clues were issued. Very well done also to the following select group, who recognised that these were the first five decimals in *Pi*: **Jim Cristiano**, director, category management, HDT Automotive Solutions LLC, Livonia, Michigan, USA: **Amparo Botella**, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; and everyone else who had a go.

SOLUTIONS

*Sudi Sudarshan*

*Here is my answer: 9*

*Solution: The numbers seem to represent the decimal portion of the value of Pi which is 3.141593 to 6 decimal places. The numbers in the sequence are: 0.1, 0.04, 0.001, 0.0005, 0.00009, ... which when added together gives 0.14159 which is the fractional portion of the value of 5 to 5 decimal places.*

*Jim Cristiano*

*We are looking at pi: 3.14159. Therefore the next number is 0.00009.*

*Amparo Botella*

*The missing decimal number is 9 = .00009. The 141592 are the decimals of pi number: 3,141592*

*Andrew Knox*

*The last single digit in this pyramid is the next digit in Pi (after the decimal place) Pi being 3.1415926.. as I remember it)*