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To solve the problem, we need to identify the pattern in the number of t-shirts Ollie sells each month and then use that pattern to predict how many shirts he will sell in August.

Step 1: Analyze the given data

The number of shirts sold each month is as follows:
- January: 120 shirts
- February: 60 shirts
- March: 240 shirts
- April: 120 shirts
- May: 480 shirts
- June: 240 shirts

Step 2: Identify the pattern

Let's examine the sequence of numbers:
- January: 120
- February: 60
- March: 240
- April: 120
- May: 480
- June: 240

We can observe that the sequence alternates between two different patterns:
1. Odd months (January, March, May): The number of shirts sold doubles from one odd month to the next.
- January: 120
- March: 240 (double of 120)
- May: 480 (double of 240)

2. Even months (February, April, June): The number of shirts sold is half of the previous odd month.
- February: 60 (half of 120)
- April: 120 (half of 240)
- June: 240 (half of 480)

Step 3: Predict the number of shirts sold in July and August

#### July (Odd month):
- The pattern for odd months is that the number of shirts doubles from the previous odd month.
- May: 480 shirts
- Therefore, July: \( 480 \times 2 = 960 \) shirts

#### August (Even month):
- The pattern for even months is that the number of shirts is half of the previous odd month.
- July: 960 shirts
- Therefore, August: \( 960 \div 2 = 480 \) shirts

Final Answer:

Ollie will sell \(\boxed{480}\) shirts in August.