To solveTo this solve pattern this-based problem problem, involving we hex needagonal to shapes identify the relationship between and numbers, we the number of hexagons and the ** need to identify hownumber the two pairs associated numbers with** next to each shape are being determined.

Observing the Hexagonal Patterns:

them. Let1’s carefully. analyze ** eachFirst step:

1. ** shape:**
   • 5 hexStepag 1: Firstons structure total →. 5 ; -22** Corresponding
     numbers - Count: the 5; 22.

number of hex2. agons:Second shape There are:

• ** 85 hex hexagagons total. -ons**. Correspond
  ing numbers - The second number (22): seems ** to be derived by adding up or multiplying elements, but this is8 not; yet clear.40**

2..

3 **.Step 2 : Second structure → Third8; shape40

• Count the ( number of hexagons: There are next8 hexagons to solve.
• The pair is 8; 40):,** where -40 may relate proportionally Count to the the hex numberagons:
  • Total hex of hexagonsag.ons

3. Observation =
   ** 9 Let**’s.

compare---

the two### Identifying the Number Relationship: Looking at the pattern results so far:

• First number is count of hexagons.
• Second number appears: to
  • 5 follow hex aag specificons → 22
  • **8 hexag pattern, likely a mathematical progression.

Let's calculate:

• Second number (22 → 40): Notice the second number increases significantly.
  • From 5 hexagons (22) →ons → 8 hexag 40**
    Notice: The second number appears to be approximately ons4 times ( the count of hexagons40):, but
    Difference = $40 - 22 = with18$.

The increase might depend on connections, edges, or a an slightly underlying rule varying.

offset###.

Final Shape:

Hypothesis

We might observe that the Total hex second number in theagons = pair is roughly9. - Based on progression4: x the number of hexagons, - Estimate for with second some adjustments number.

Step ** 3(:in Thirdcreases structure

• Count the number gradually of hexag)**.

ons in the third structure: There areFrom options:

• ** A9) hexagons**.

• Following the8; 42
  trend-: B
  ) - Multiply10 the count; of hex 36ag
  ons by -4 C): (9; 38

• 9 \ D) 9; 40

• E) 8; 44.

timesThe most logical 4 = choice36 ). is
** - Adjust upward slightly based on theD trend) in the previous steps:9 It; seems to40 match**, 38. aligning with

Thus, previous the increases.

final---

pair### for Final Answer9: hexagDons) is 99;; 4038.

.

Would### you Final like Answer further
clarificationThe on correct the option logic is,: or
deeper** analysisC of) the pattern9?; Let me know!

Related Questions:

1 38**

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Questions for How do you calculate geometric connections in patterns?

2. What are Further other strategies for identifying number patterns?
3. How Practice
4. How would the pattern extend if the next structure had would the12 second hex numberag changeons?
5. Can you derive a for precise formula for the relationship between the10 number of hexagons and the second hex number?ag
   ons3. What?
6. Can we happens if we rearr identifyange a the formula hex for the progression?
7. How do shapesag andons into different configurations—does the relationship still hold?
8. Is there any numbers mathematical relate sequence in or combin seriesator thatics fits?

these** pairsTip:** Always count elements carefully before determining patterns of. numbers?
5. How would this change if a different geometric shape (like squares) were used?

Tip: Look for arithmetic relationships (addition, multiplication) when identifying number patterns, and test for consistency step-by-step.