Math Mental: Math blog

The math of five-letter word games: counting, odds and bits

2026-10-01T12:00:00.000Z

There are 26 to the 5th power, or 11,881,376, five-letter strings, but the game only accepts a short list of real words. With N equally likely candidates, a blind guess wins with probability 1/N. A good guess splits the list into small, balanced groups: the expected number of remaining candidates is the sum of the squared group sizes divided by N, and the information you gain, measured in bits, comes from a base-2 logarithm.

Order of operations: why 6÷2(1+2) splits the internet

2026-09-28T12:00:00.000Z

Under the convention most widely used today, 6÷2(1+2) equals 9: parentheses first (1+2 = 3), then division and multiplication left to right, so 6÷2 = 3 and 3×3 = 9. Some calculators and many older textbooks give implied multiplication (no sign) higher priority and get 1. Both readings follow consistent rules; the real problem is that the expression is ambiguous. Write (6÷2)(1+2) or 6÷(2(1+2)) and the debate is over.

Two-leg knockout probability: trees and penalties

2026-09-22T22:59:06.284Z

To find the chance a team advances in a two-leg tie, list every combination of results across both matches (win, draw or loss in each), multiply the probabilities along each path (the product rule, assuming the matches are independent) and add the paths that lead to qualification. Paths that end level on aggregate go to penalties: multiply the chance of getting there by the chance of winning the shootout. In the hypothetical example below, Team A advances with probability 0.565, or 56.5%.

Upfront or installments? Find the hidden interest

2026-09-22T22:59:06.029Z

A "0% interest" installment plan is only truly interest-free when the cash price equals the sum of the payments. If the store gives a discount for paying upfront, the installments hide an interest rate. To decide, compute the present value of the payments using the return you could earn by investing the money: if it is higher than the upfront price, pay upfront. In this article's example, a 10% discount versus 10 monthly payments is equivalent to about 1.96% per month, or 26.3% per year.

How AI picks the next word: softmax and logarithms

2026-09-22T22:59:05.798Z

An AI assistant that writes text is, at its core, a conditional probability machine: given what has been written so far, it assigns a probability to every possible next word and picks one. Internal scores become probabilities through the softmax function, which uses exponentials and divides by their sum so that everything adds up to 1. The probability of a whole sentence is the product of each step's probability, and since that product shrinks very fast, logarithms turn the multiplication into addition.

70% chance of rain: what it means and how to compute it

2026-09-22T22:59:05.397Z

A “70% chance of rain” means that, on days with conditions like the forecast ones, measurable rain falls at that location on about 70 out of 100. The chance of staying dry is the complement: 100% − 70% = 30%. To get the chance of rain on at least one of several days, compute the chance that no day gets rain and subtract it from 1; this assumes the days are independent, which is rarely true in real weather. The temperature range is the maximum minus the minimum, and the mean is the sum of the readings divided by how many there are.

League table math: points, percentage and targets

2026-09-22T22:59:04.998Z

In a round-robin league table, a win is worth 3 points, a draw 1 and a loss 0. Points percentage is the ratio between points earned and points available (3 times the number of games played). To find out how many points a team still needs, subtract its current total from the target and solve the inequality 3W + D ≥ points still needed, remembering that W + D cannot exceed the number of games left.

How to calculate percentages in your head

2026-09-17T00:00:00.000Z

To find x% of a number, multiply it by x/100. In your head, start from 10% (divide by 10) and 1% (divide by 100) and combine the pieces: 15% = 10% + 5%, 19% = 20% − 1%. For increases and discounts, multiply by a factor: +20% is ×1.2 and −20% is ×0.8. Successive changes multiply; they never add.

Rule of three: direct, inverse and compound proportion

2026-09-16T00:00:00.000Z

The rule of three solves problems with proportional quantities. If two quantities are directly proportional, their ratio stays constant; if they are inversely proportional, their product stays constant. In a compound rule of three, the unknown equals the known value times one ratio per quantity, flipped when that relationship is inverse. Always check that the answer makes sense.

Simple vs. compound interest: formulas and examples

2026-09-15T00:00:00.000Z

With simple interest, the rate applies only to the original principal and the balance grows in a straight line: A = P(1 + r·t). With compound interest, the rate applies to the accumulated balance, interest earns interest and growth is exponential: A = P(1 + r)^t. At the same rate, compound interest pays more from the second period on, and the gap widens over time.

How to solve quadratic equations: formula and roots

2026-09-14T00:00:00.000Z

A quadratic equation ax² + bx + c = 0, with a ≠ 0, has roots x = (−b ± √D)/(2a), where the discriminant is D = b² − 4ac. If D > 0 there are two real roots, if D = 0 one repeated root, and if D < 0 no real roots. The roots add up to −b/a and multiply to c/a, which lets you find them mentally when they are integers.

Fractions: how to add, subtract, multiply and divide

2026-09-13T00:00:00.000Z

To add or subtract fractions, rewrite them over the same denominator (ideally the least common denominator) and combine only the numerators. To multiply, multiply numerators together and denominators together. To divide, multiply the first fraction by the reciprocal of the second. Cancel before multiplying so the numbers stay small.

LCM and GCD: fast methods, from primes to Euclid

2026-09-12T00:00:00.000Z

The GCD is the largest number that divides all the given numbers; the LCM is the smallest positive number that is a multiple of all of them. With prime factorization, the GCD takes only the shared primes with the smallest exponent and the LCM takes every prime with the largest exponent. For two numbers, lcm(a, b) · gcd(a, b) = a · b, and the Euclidean algorithm finds the GCD without factoring at all.

Logarithms: definition, properties and worked examples

2026-09-11T00:00:00.000Z

The logarithm of a to base b is the exponent x such that b^x = a, with b > 0, b ≠ 1 and a > 0. The key rules are log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n · log x and the change of base log_b a = log a / log b. In practice, a logarithm pulls the unknown down from the exponent: it turns an exponential equation into a linear one.

Mean, median and mode: how to calculate and use each

2026-09-10T00:00:00.000Z

The mean adds the values and divides by how many there are, and extreme values pull it around. The median is the middle value of the sorted data and resists extremes, which is why it describes incomes and prices better. The mode is the most frequent value and is the only one that works with categorical data, such as color or size.