# What is the missing in this pattern?

there. I came across a problem about number patterns as described in the following picture. This puzzle is from a workbook for primary school students who intend to sit the exams for enrollment in selective high schools in Sydney.

The numbers in each hexagon must be related to each other somehow and the numbers in the neighouring areas may be related to each other in a way.

But I can't find out the relations except for the center areas. Would you like to give it a go and find out the relationships? Thank you.

• lol I failed to solve it but according to my logic it is either 119 or 154. so it should be 119... according to me which looks more suitable :D .... how i solve it well there is nothing mathematical answer just look middle 2 hexagon if you see the number in front of 6 is lesser number than number in front of 3 and number right side of hexagon is lesser number from number of left side hexagon. so it is on left side so it must be greater number. either 119 or 154. now 67 is odd number too so it must be odd number 119 is odd. i – Sayed Mohd Ali Sep 2 '19 at 13:58
• Thank you, Sayed. But I'm not quite clear about your explanation. Would you please explain it in a clearer way? Thanks again. – Michael May Sep 2 '19 at 23:33
• it is not something I came up with proper answer and explaination, it is just assumption on the basis of hexagons... if you see the top and down hexagon they look just like mirror to each other that means they both are affected by left and right hexagon someway. so according to the mirror theory it should be a larger number. just check the top hexagon and compare the numbers with the down hexagon. you will able to see one thing the number in front of 6 is smaller number as compare to number in front of 3. – Sayed Mohd Ali Sep 3 '19 at 6:07
• in both top and down hexagon. the number next to 3 is smaller number as compared to the number next to 6 is larger number. so here the number is next to 6 so it should be larger number than 22 and also the difference is large so it must be either 119 or 154. – Sayed Mohd Ali Sep 3 '19 at 6:07
• Thank you, Sayed. Your explanation is clearer this time and I think I can understand what you mean. Maybe this problem doesn't have a precisely rational answer and leaves space for multiple interpretation. Thanks very much, anyway. – Michael May Sep 3 '19 at 7:26

Starting from the 1 you can easily see the series 1, 2, 4, 7, 11, 16... ($$a_0 = 1, a_n = a_{n-1} + n$$) if you move clockwise along the outer vertices of the figure, skipping one. This works until you get to 37, where the next move would get you back to 1, but the next number on the series would be 46 which is next to it. Continuing the series from 46 the pattern shows up again (46, 56, 67, 79...). When you arrive at the question mark, the number that would follow 137 would have to be 137 + 17 = 154.