How to get total of 67 through calculation from six of three
The correct answer and explanation is:
It looks like you want to find a way to get a total of 67 through a calculation involving six numbers, each being the number three (3). In other words, you want to use six 3s combined with mathematical operations to make 67.
Step 1: Understand the problem
You have six 3s:
3, 3, 3, 3, 3, 3
Your goal: Use any combination of operations (+, -, ×, ÷, powers, concatenation, parentheses) to get 67.
Step 2: Try simple arithmetic combinations
- Sum of six 3s:
3 + 3 + 3 + 3 + 3 + 3 = 18 (too small) - Multiplying all:
3 × 3 × 3 × 3 × 3 × 3 = 729 (too large) - Using powers and concatenation:
One clever way is to create larger numbers by concatenating 3s, such as “33” or “333”.
Step 3: Constructing 67 from six 3s
Let’s try:
- Use “33” (two 3s)
- Use “3” (one 3)
- Use “3” (one 3)
- Use “3” (one 3)
- Use “3” (one 3)
We still need to count how many 3s we have.
“33” uses 2 threes. Then 3 + 3 + 3 + 3 uses 4 threes, total 6.
Try:
33 + 3 × 11 = ?
We need 11 from three 3s. How?
- 3 + 3 + 3 + 3 + 3 + 3 = 18 (too big for 11)
Try another approach:
Try this:
(33) + (33) + (3/3) = ?
Count the 3s:
33 uses 2 threes, twice = 4 threes
3/3 uses 2 threes
Total = 6 threes
Calculate:
33 + 33 + (3/3) = 33 + 33 + 1 = 67
Final answer:
33+33+33=6733 + 33 + \frac{3}{3} = 67
Explanation (300 words)
To reach the total of 67 using exactly six of the number 3, we need to creatively use the digits and arithmetic operations. First, understand that simply adding six 3s (3 + 3 + 3 + 3 + 3 + 3) gives only 18, which is far from 67. Multiplying all six 3s (3 × 3 × 3 × 3 × 3 × 3) results in 729, which is much larger than 67. So simple addition or multiplication won’t work.
A helpful trick is concatenation: combining two or more 3s to form a larger number like 33 or 333. Here, we use “33” twice. Each “33” uses two 3s, so two “33”s use four 3s in total. This leaves two more 3s to reach six 3s in total.
Now, to reach exactly 67, add the two 33s:
33 + 33 = 66
You need just 1 more to get 67. Using the last two 3s, divide one by the other:
3 ÷ 3 = 1
Finally, add the 1 to 66 to get 67. This way, we have:
33 + 33 + (3/3) = 67
This solution uses exactly six 3s, combines concatenation, addition, and division. It’s a neat demonstration of how flexible mathematical operations and digit manipulation can be used to reach a target number. This approach is commonly seen in number puzzles where the goal is to form numbers using limited digits and standard arithmetic operations.