Satt, stundum, ósatt: always, sometimes or never true?
Satt, stundum, ósatt means true, sometimes, false. Ten levels of algebra statements in Icelandic, from 3 + 4 = 7 to (a + b)² = a² + b², where you decide whether each one is always true, sometimes true or never true.
A judgement instead of an answer
Most algebra exercises ask you to simplify or to solve. This game asks something else: is the statement always true, sometimes true or never true? That sounds easier than solving, and it is not, because it is exactly where the classic mistakes live. Someone who believes that (a + b)² is a² + b², or that a minus in front of a bracket changes only the first term, will answer those statements with confidence and get them wrong.
Every answer comes with its evidence. One number that makes the two sides differ is a counterexample, and a single counterexample rules out always. One number that makes them equal rules out never. Showing that something is always true takes a rule, such as the distributive law, and the explanations name the rule that settles it.
Try numbers on the balance
Under each statement there is a small lab. You set each letter with plus and minus buttons or a slider, and both sides are worked out as you go, on the same balance scale that Vogin uses. The beam stays level when the two sides are equal and tips towards the larger side when they are not. If a side does not exist for the chosen numbers, because it divides by zero or takes the square root of a negative number, the scale fades and that choice does not count.
After a wrong answer the lab opens by itself with an example already on the scale, next to the explanation, so the mistake is shown and not just marked wrong.
The ten levels
- Jafnaðarmerkið. Numbers only, so every statement is simply true or false.
- Bókstafur er tala. x can be any number, and the three answers appear.
- Sams konar liðir. Like terms, the ones that can be added together.
- Mínus og svigar. A minus in front of a bracket reaches every term inside it.
- Veldi og formerki. Why −x² and (−x)² are not the same.
- Hvað dreifist? Multiplication distributes over a sum, powers do not.
- Kvaðratrætur. A square root does not split over a sum.
- Brot. Factors can be cancelled, terms never.
- Jöfnur og samsemdir. Sometimes true means the equation has a solution.
- Allt blandað. Everything mixed.
Each level has six statements and five correct answers open the next one. A short reference card opens at the start of each level and is one tap away after that. A new idea is never used before the level that introduces it. Statements you get wrong come back in later rounds.
Surprisingly sometimes
The most useful statements are the ones that are neither always nor never true. √(a + b) = √a + √b is false for a = 9 and b = 16, but true whenever a or b is 0. x² = x holds for exactly two numbers, 0 and 1, and dividing both sides by x loses one of them. a + b = ab holds when a and b are both 2. None of them can be answered by remembering a rule halfway; you have to try numbers.
Icelandic maths vocabulary
| Icelandic | English | In the game |
|---|---|---|
| fullyrðing | statement | what an equals sign makes: a claim that is true or false |
| jafna | equation | a statement with an equals sign and letters |
| samsemd | identity | an equation that is true for every number |
| lausn | solution | a number that makes an equation true |
| gagndæmi | counterexample | one is enough to rule out always |
| liður | term | 3x and 2 in 3x + 2 |
| stuðull | coefficient | 3 in 3x |
| þáttur | factor | 2 and x + 1 in 2(x + 1) |
As in Vogin, the pair worth learning is liður and þáttur. The fraction level is built on that difference: (2x + 4)/2 = x + 2 is always true, while (3x + 1)/3 = x + 1 is never true, because the 3 in 3x belongs to one term and not to the whole numerator, so it cannot be cancelled.
What is the difference between an equation and an identity?
An identity is an equation that holds for every value of its letters. Other equations hold for some values, or for none. 2(x + 1) = 2x + 2 is an identity: expand the bracket and the two sides match. 2x + 1 = 7 holds only when x is 3, which is its solution. x + 1 = x has no solution, since no number is one more than itself.
Satt, stundum, ósatt, Icelandic for true, sometimes, false, is built on those three kinds. Each statement is always, sometimes or never true, and you decide which. More than a hundred of them run across ten levels, from the equals sign to powers, square roots and fractions, and a program checks the class of every one. Some are surprising: √(a + b) = √a + √b is sometimes true, because it holds whenever a or b is 0. A balance scale lets you try numbers, and every answer comes with the example or the rule that settles it.
Frequently asked questions
- Is (a + b)² equal to a² + b²?
- Only sometimes. Expanding gives (a + b)² = a² + 2ab + b², and the middle term 2ab disappears only when a or b is 0. With a = 2 and b = 3 the left side is 25 and the right side is 13, and that one counterexample is enough to show the statement is not always true.
- Why do examples never prove that a statement is always true?
- Because the next number could be the one that fails. The statement a + b = ab works when a and b are both 2, and fails when they are 1 and 3. To know that something holds for every number you need a rule, such as the distributive law, and the explanations name the rule that settles it.
- Do I need Icelandic to play it?
- Not for the algebra, which is written the same way everywhere. The three answers are Alltaf satt, Stundum satt and Aldrei satt: always, sometimes and never true. The explanations are in Icelandic and keep returning to a small set of maths words, so a learner picks up liður, þáttur, lausn and gagndæmi along the way.