How to use this quiz: choose an answer before opening the explanation. Each answer is supported by two independent sources checked on 2026-08-27.
10 mathematical patterns questions
1 · Warm-up
When two fair six-sided dice are rolled, which total can be produced by the most ordered outcomes?
- 5
- 6
- 7
- 8
Show the answer, explanation, and sources
Answer: 7. Seven has six of the 36 possible ordered outcomes.
The pairs 1+6 through 6+1 all total 7, giving six of the 36 equally likely outcomes. No other total has as many routes.
Sources: Portland Community College and University of California, Berkeley.
2 · Warm-up
Which even number has exactly two positive divisors?
- 0
- 4
- 6
- 2
Show the answer, explanation, and sources
Answer: 2. Two is the only even prime number.
A prime has exactly two positive divisors: 1 and itself. Every larger even number is divisible by 2 as well, so it cannot be prime.
Sources: Wolfram MathWorld and The Open University.
3 · Core
Ignoring leap day, about how many people make a shared birthday more likely than not?
- 23
- 50
- 100
- 183
Show the answer, explanation, and sources
Answer: 23. A group of 23 people contains 253 possible pairs.
The probability depends on comparing every possible pair, not one person with the rest of the group. With 23 people, 253 pairings place the chance of at least one match just above 50 percent.
Sources: Scientific American and University of Illinois.
4 · Core
A paper strip is joined into a loop after one half-twist. How many sides and edges does it have?
- Two sides, two edges
- One side, one edge
- One side, two edges
- Two sides, one edge
Show the answer, explanation, and sources
Answer: One side, one edge. A Möbius strip has one continuous side and one continuous edge.
Tracing the surface without lifting a finger returns to the starting point after covering what initially appeared to be both sides. The boundary forms one continuous edge as well.
Sources: Smithsonian Institution and MIT OpenCourseWare.
5 · Core
On the real number line, where does the repeating decimal 0.999… land?
- Just below 1
- One infinitesimal below 1
- Exactly at 1
- Just above 1
Show the answer, explanation, and sources
Answer: Exactly at 1. 0.999… is exactly equal to 1.
Let x = 0.999…. Then 10x = 9.999…, so subtracting x gives 9x = 9 and x = 1. The two notations name the same real number.
Sources: Harvey Mudd College and University of California, Berkeley.
6 · Core
Who supplied mathematician Edward Kasner with the word “googol”?
- Kasner himself
- A newspaper editor
- A chess champion
- His nine-year-old nephew
Show the answer, explanation, and sources
Answer: His nine-year-old nephew. Kasner's nephew Milton Sirotta proposed the word “googol.”
Kasner asked his nine-year-old nephew Milton Sirotta to name 10¹⁰⁰. Milton proposed “googol,” a word that later inspired the spelling of Google.
Sources: Guinness World Records and Wolfram MathWorld.
7 · Core
A flat map needs at most how many colors so regions sharing a border never match?
- 4
- 5
- 6
- 7
Show the answer, explanation, and sources
Answer: 4. Four colors are sufficient for every planar map.
The four-color theorem guarantees four colors are enough when regions that share a boundary must differ. Merely touching at a corner does not count.
Sources: Smithsonian National Museum of American History and American Mathematical Society.
8 · Core
One taxi number equals both 12³ + 1³ and 10³ + 9³. Which number is it?
- 1728
- 1729
- 1730
- 2197
Show the answer, explanation, and sources
Answer: 1729. 1729 is the Hardy–Ramanujan taxicab number.
Hardy called his taxi number dull; Ramanujan immediately noted it is the smallest positive integer expressible as two positive cubes in two different ways.
Sources: American Mathematical Society and University of Houston.
9 · Challenge
What result does the subtraction 7641 − 1467 produce?
- 4356
- 6000
- 6174
- 7641
Show the answer, explanation, and sources
Answer: 6174. 7641 − 1467 equals 6174, the fixed point of Kaprekar's routine.
Because 7641 − 1467 equals 6174, repeating the routine produces the same result. Every four-digit starting value with at least two distinct digits reaches 6174 within seven rounds.
Sources: University of St Andrews and Berkeley Math Circle.
10 · Challenge
In many datasets spanning several scales, which leading digit appears about 30 percent of the time?
- 9
- 7
- 5
- 1
Show the answer, explanation, and sources
Answer: 1. Benford's law predicts that 1 appears first about 30 percent of the time.
For suitable naturally occurring datasets, 1 leads about 30.1 percent of values, while 9 leads only about 4.6 percent. The pattern is logarithmic, not uniform.
Sources: National Institute of Standards and Technology and Carnegie Mellon University Software Engineering Institute.
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