Challenging math puzzles and brain teasers involving basic and ... - Free Printable

Final Answer:
The numbers in the left box are all powers of 2:
4 = 2², 8 = 2³, 16 = 2⁴, 32 = 2⁵, 36 = ? Wait — 36 is not a power of 2 (2⁵ = 32, 2⁶ = 64), and 64 = 2⁶. So left box has: 4, 8, 16, 32, 64 — those are powers of 2 from 2² to 2⁶ — that’s five numbers. But there are six circles: 4, 8, 16, 32, 36, 64. So 36 is the odd one out.

Right box: 25 = 5², 32 = 2⁵, 36 = 6², 49 = 7², 64 = 8². So most are perfect squares: 25, 36, 49, 64 — that’s four squares; 32 is not a square.

But maybe the task is to find which number appears in both boxes? Let's list:

Left box numbers: 4, 8, 16, 32, 36, 64
Right box numbers: 25, 32, 36, 49, 64

Common numbers: 32, 36, 64 → three numbers.

Wait — perhaps the problem is to find the number that is *only* in one box? Or maybe it's a pattern: both boxes contain numbers that are either perfect squares or powers of 2 — but we need to find which number doesn’t belong in its box.

Let me re-express all numbers with prime factorizations:

Left:
- 4 = 2² ✔ power of 2, also square
- 8 = 2³ ✔ power of 2
- 16 = 2⁴ ✔ power of 2, square
- 32 = 2⁵ ✔ power of 2
- 36 = 2² × 3² = (2×3)² = 6² ✔ perfect square, but *not* a power of 2
- 64 = 2⁶ ✔ power of 2, also square

So left box has five powers of 2 and one perfect square that’s not a power of 2 (36).

Right box:
- 25 = 5² ✔ square
- 32 = 2⁵ ✔ power of 2 (not square)
- 36 = 6² ✔ square
- 49 = 7² ✔ square
- 64 = 8² = 2⁶ ✔ square and power of 2

So right box has four perfect squares (25, 36, 49, 64) and one power of 2 that’s not a square (32).

Now, what could the question be? Since no explicit question is given, but this is a common type: “Which number does NOT belong?” or “Find the number that appears in both sets” — but the instruction says “Solve the problem accurately”, implying there is a standard interpretation.

Looking at typical puzzles like this: two sets, find the number that is the *only* one satisfying a certain property across both.

Another idea: maybe each box contains numbers with exactly 3 positive divisors? No — numbers with exactly 3 divisors are squares of primes: 4 (2²), 9, 25, 49 — left has 4, right has 25, 49 — but 36 has 9 divisors, 64 has 7 divisors, etc.

Wait — let’s count how many numbers are perfect squares:

Left: 4, 16, 36, 64 → 4 squares
Right: 25, 36, 49, 64 → 4 squares
Common squares: 36, 64
Also 4 and 16 only left; 25 and 49 only right.

But again — without the exact question, we must infer from standard format.

Hold on — perhaps the task is: “Which number is in both boxes?” That’s a common simple question for such diagrams.

Intersection = {32, 36, 64}

But usually such puzzles expect a *single* answer. So maybe only one is truly common *and* satisfies a condition.

Check if any number appears exactly once across both boxes? No, all are unique except those three.

Wait — maybe the real task is: “One of the numbers is misplaced. Which one?” That fits educational homework.

In left box: five numbers are powers of 2 (4, 8, 16, 32, 64). 36 is not. So 36 is the odd one out in left.

In right box: four numbers are perfect squares (25, 36, 49, 64). 32 is not. So 32 is odd one out in right.

But which box is “correct”? Perhaps the intended rule is: each box should contain only perfect squares. Then left box has 4, 16, 36, 64 as squares; 8 and 32 are not squares → two errors.

Alternatively: each box should contain numbers that are either squares or powers of 2 — then all fit.

Wait — let me search my memory: This exact diagram appears in some IQ/test questions where the question is: “Which number is the only one that is a perfect square and a power of 2?”

Numbers that are both perfect squares and powers of 2 are powers of 2 where exponent is even: 2⁰=1, 2²=4, 2⁴=16, 2⁶=64, 2⁸=256, etc.

So in left box: 4, 16, 64 qualify.
In right box: 64 qualifies. 36 is square but not power of 2; 25, 49 same; 32 is power of 2 but not square.

So 64 is the only number that is in both boxes *and* is both a perfect square and a power of 2.

But still, why would that be the answer?

Alternatively, maybe the problem is simply: List the numbers that appear in both boxes.

But the instruction says “Solve the problem accurately” and “provide only the final answer”.

Given typical multiple-choice style hidden in such images, and since the user didn’t specify the question, but in many standardized practice sheets, this image is paired with:
“Which number appears in both groups?”
And the expected answer is the intersection — but they usually expect a single number. However, there are three.

Wait — double-check the image description:
Left box: circles with 4, 8, 16, 32, 36, 64 → 6 numbers
Right box: 25, 64, 32, 49, 36 → 5 numbers

Yes, 32, 36, 64 appear in both.

But maybe the original problem says: “One number is repeated by mistake. Which one shouldn’t be there?” That would be ambiguous.

Let me think differently: Could it be about sum?

Left sum: 4+8+16+32+36+64 =
4+8=12, +16=28, +32=60, +36=96, +64=160
Right sum: 25+64+32+49+36 =
25+64=89, +32=121, +49=170, +36=206

No obvious relation.

Product? No.

Number of letters in spelling? Unlikely.

Wait — perhaps the actual question (implied by format) is: “What is the missing number?” but there is no blank.

Given the constraints, and that the assistant must provide a single final answer, and this is a known puzzle: the intended answer is 64, because it is the only number that is both a perfect square and a power of 2 *and* appears in both boxes — while 32 is power of 2 only, 36 is square only.

In many curricula, they teach that numbers like 64 are special: 8² and 2⁶.

Moreover, if you sort each box:
Left: 4, 8, 16, 32, 36, 64 → powers of 2 except 36
Right: 25, 32, 36, 49, 64 → squares except 32

The only number that fits *both* primary patterns (power of 2 and square) is 64.

I think the expected answer is 64.

Let me verify once more:
- Powers of 2 ≤ 100: 1, 2, 4, 8, 16, 32, 64
- Perfect squares ≤ 100: 1, 4, 9, 16, 25, 36, 49, 64, 81
Intersection: 1, 4, 16, 64
From the two boxes, which of these appear? 4 (left only), 16 (left only), 64 (both), 1 (neither). So only 64 is in both boxes *and* in the intersection of powers of 2 and squares.

Thus, if the question is “Which number is common to both boxes and is both a perfect square and a power of 2?”, answer is 64.

Since no question is stated, but the only logically unique answer is 64, I will go with that.

Final Answer: 64