Which of these numbers is both rectangular and triangular? Explain 1,5,21?

Answer 1

My understanding or rectangular numbers is they are numbers that can be formed using n rows and n+1 columns.

The rectangular numbers are:
1 x 2 = 2
2 x 3 = 6
3 x 4 = 12
4 x 5 = 20
etc.

Using this definition, none of your numbers would be rectangular.

However, if we assume they meant to use the looser definition of rectangular numbers, they could mean numbers that can be factored as two numbers (greater than 1). In this case 21 = 3 x 7 (3 rows, 7 columns)

Rectangular numbers are:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, etc.

Triangular numbers are:
1, 3, 6, 10, 15, 21, etc.

Imagine 1 on the first row, 2 on the next row, 3 on the third row, 4 on the fourth row, etc.

1+2+3+4+5+6 = 21

So I think the intended answer is 21.

Answer 2

Two contributors have given 21 which is correct in that it is a triangular and a rectangular number.

However, I wonder if there is a trick in the question. One (1) is often represented as the first triangular number and by the same reasoning 1 would be a rectangular number.

I suspect that some will argue that it is and others will argue it isn't! This is reminiscent of the old debate 'is 1 a prime number?'
I suspect

Answer 3

Triangle numbers.Numbers of snooker balls you can get in the triangular set up at the start of a game.
If n = the number of rows then B(the number of balls) is given by B =((n+1)/2)) x n giving the series 1,3,6,10,15,21,28.....
21 is both triangular and rectangular.
neither of the others are.

Answer 4

21 could be considered triangular because it divides exactly by 3 but 1 and 5 could be rectangular because it doesn't divide by 4 and 12 would be a square.

Answer 5

21

a number is rectangular when it has two factors that are greater than 1 so x * y = n (x,y not equal to 1)

1.......................= 1
1+2...................=3
1+2+3...............=6
1+2+3+4..........=10
1+2+3+4+5......=15
1+2+3+4+5+6..=21

a number is triangular when it follows the above pattern

Answer 6

According to research it is true that triangular sandwiches taste better than square ones, though I don't agree, nothing like a bacon and egg sarnie cut square.

Answer 7

21

rectangular just means it can be put into rows and columns

* * * * * * *
* * * * * * *
* * * * * * *

so there are 21 stars in a rectangle of 3x7

you can also arrange them in a triangle

*
**
***
****
*****
******

again 21 stars with one extra on each line as you go down

e.g. 5 can't be triangle

*
**
**

as you have 2 stars only on the second and third lines

and 1 can't be a triangle as you have to have at least 3 points to make a triangle

Answer 8

21, get 21 match stick and try.

Answer 9

please explain the question