Logic Puzzle tasks and deductions

Use negative rules for maximum impact. Whenever a rule includes not, never or cannot, turn it to a positive. This is a skill you will want to practise and refine with the Decision Making questions as you get ready for Test Day.

Use limitations to your advantage with a negative rule. If there are five people in the puzzle and you are told a detail that does not apply to two of them, then there are only three people it could be.

Logic Puzzles with an algebra task can take a range of formats:

• You may see word problems where you must solve algebraically for an unknown value. These may feel similar to maths questions in other exams you have taken.
• You could have a question using symbols or shapes to represent numerical values in a system of equations. These formats may be very unfamiliar, but the symbols or shapes are simply variables. You can solve for them as if they were letters in conventional algebra.

As with any Logic Puzzle, you must keep an eye on the goal when the task is algebra. This means you need to consider whether the correct answer is:

• A symbol or shape, so you must solve for the unknown variable.
• A number, implying you must count or calculate a precise value.
• A name, word or phrase, so you must figure out which person, thing or category answers the question.
• Something that must be true, so you can potentially disprove the wrong answers as an alternative to the straightforward proof of the correct answer.

In an algebra task, use the goal and the set-up to guide your deductions.

• Identify the first step you must take to work towards the goal.
• In a word problem, you will likely just set up and solve for the unknown.
• In other formats, figure out which equation is simplest to use to make a deduction.
• Try to proceed in some sort of order so that one deduction will let you work with remaining rules or equations to make another deduction.
• Continue in this approach until you use all the rules and equations, or until you eliminate three wrong answers.

In a pure sequencing task, the order is in one dimension. This allows you to use a row or column of blanks in your sketch, which you can then fill in with abbreviations, such as the people’s initials.

Sometimes, a Logic Puzzle will combine sequencing with another task:

• If the task is sequencing and spatial: Let the spatial element drive your sketch, as you will have to think in more than one dimension to make deductions.
• If the task is sequencing and matching: Use a fixed order to orientate your sketch. Most often, this will be numbers, such as prices of items or 1st through 5th place in a race. You can then arrange people or things according to higher or lower numbers.

Note that puzzles with sequencing and matching tasks are extremely common – you are likely to see one on Test Day – whilst puzzles with sequencing and spatial tasks are extremely rare.

Most Logic Puzzles will include a list of rules you can attack in any order. The quickest approach is to start with the most concrete rule. Train yourself to look for the rule (or the part of a very long rule) that is the most specific, fixed or tangible and use it first.

• Any rule that assigns a person or thing to a fixed position is the most concrete. You can add this rule directly into a sketch.
• A rule that gives relative positions is a bit less concrete. You can notate the people or things in relation to each other, but you may need more information before you can add them into the sketch.
• Negative rules are generally quite a bit less concrete. Try to turn a negative rule into a positive statement; sometimes, you can deduce a fixed or relative position, or you may need to make a notation to combine with other deductions.
• Complex rules that include or exclude multiple possibilities are generally the least concrete. It’s very hard to work with these rules until you have made some notations – ideally, some deductions – from the other rules.

As you practise, train your eyes to pick out rules that are simpler to start with. This will be challenging at first, but it’s a skill you can learn. This way, you can avoid reading all the rules in detail until it is time to untangle the logic and make deductions.

Some spatial tasks will include maps or diagrams. Sometimes, these are straightforward to use on the screen, so you can work quickly to make deductions.

If there isn’t a diagram, or the diagram is not ideal, you may prefer to draw a diagram in your notebook. This can be a quick, rough sketch – it’s merely a tool to aid your thinking.

Many Logic Puzzles will include negative rules that tell you what cannot be true. These rules are a bit tricky to work with at first glance – it can be hard to align these impossibilities as you try to deduce what must be true.

Try to turn negatives into positives. This will take some practice. It will also generally be simpler once you have used the more concrete rules or made a deduction or two. The more you fill in the exact slots or matches, the fewer possibilities there are for negatives to become positives.

Use the limited options to help see what is possible or certain based on a negative rule. There are only so many people, things, colours or positions in a Logic Puzzle.

Knowing that one entity cannot fit with one or two others could severely limit what is possible. In some cases, it may be limited to a single option, particularly in combination with other rules and deductions.

Sometimes, there isn’t a rule that assigns an entity to a fixed position. In such cases, it may be best to turn a negative rule into a positive and work from there.

Watch out for unknowns. An unknown is an entity you cannot assign to an exact position or match with anything else.

Not all Logic Puzzles have unknowns. You will notice them when they are present because:

• The set-up includes a person, thing or colour among the list of entities.
• That person, thing or colour is not mentioned in any of the rules.
• You know the unknown must go somewhere, but you cannot make any deductions about it.

You will usually have to make all other deductions, then you can see where the unknown fits in. Often, there is one remaining slot (in sequencing or spatial tasks), or one unmatched option (in matching tasks).

Matching is a very common task in Logic Puzzles. You must pair up entities from different categories.

In its simplest form, you will match up two categories of entities.

Examples of simple matching tasks in Logic Puzzles:

• each vehicle with its colour
• each person with a different activity
• each friend with a hotel room

You are likely to see Logic Puzzles that mix in another task along with matching the two categories. This combined task makes for more complexity as you work through the deductions.

The other task could be sequencing or spatial. It could also be a third category of entities, sometimes known as ‘double matching’ since each entity pairs up with two others.

Examples of complex matching tasks in Logic Puzzles:

• each vehicle of a different type and colour (double matching)
• each person with a different activity and its price (matching plus sequencing)
• each friend with a hotel room in a diagram (matching plus spatial)

In official UCAT materials, double matching is far more common than simple matching. This means there is a good chance you will have to match people or things with two other categories.

However, simple matching is often a component of other Logic Puzzles, frequently mixed with sequencing as a combined task. Puzzles with a combined sequencing and matching task are by far the most common. Double matching is also extremely common, although – despite its complexity – this is technically a ‘pure’ matching task.

For now, let’s consider the basic techniques and skills common to matching tasks. They will serve you well in any such Logic Puzzle, regardless of complexity.

In a matching task, you must decide how to focus your deductions. This means that you will usually want to match people or things with other categories, like colours or activities.

You can match across categories any way you like, and some rules may directly link secondary characters. But you will generally want to focus your work on:

• a spatial or sequencing task that can orientate your sketch,
• people or (in the absence of people) large things, like cars or houses, in the absence of a spatial or sequencing task,
• matching up secondary characteristics to the people, large things or the spatial or sequencing task.

This aligns with the everyday logic of thinking of a car’s colour or what activity a person is doing, so you will likely find it faster to start with the cars (matching colours to cars) or people (matching activities to people) instead of the other way round.

Logic Puzzles will often include one or more paired rules.

A paired rule might:

• limit the possibilities to multiple options – one of which must be true.
• exclude multiple options – none of which cannot be true.
• mix positives and negatives – you must read carefully to see what is possible/certain and what is impossible.

Be ready to turn negatives into positives. You may need to apply entities not mentioned in the rule to list what is possible, based on what the rule excludes.

Paired rules are all about elimination – they let you cross out entities that cannot match with those in the rule. You will likely need to combine rules to see what must be true.

In a matching task with several paired rules:

• Start with the most positive rule – the one that limits things to definite or possible options. Notate it first. Ideally, you will be able to pair up entities in a concrete way.
• Continue with the next most positive rule and notate it next.
• Leave the most negative rule – the one that is all or mostly a list of what cannot be true. Try to turn it into a notation of what could be true.
• Then, compare your notations to see if you can make any matches. Once you have the positive possibilities, you may see that certain matches are the only option for the remaining entities.
• Once the deductions begin, continue through your notations and cross out the possibilities you have already matched up.

As always, keep an eye on the answers as you will likely disprove wrong answers with your initial work. You may find that you prove the correct answer, even if you cannot match everything definitively.

In a Logic Puzzle that is purely about matching – whether single or double matching – beware that it may be impossible to match up everything with 100% certainty.

Sometimes, you can complete the task for all entities. The correct answer may depend on you doing so.

Other times, partial work is all you need. The correct answer may require just a few basic deductions.

Take care not to try to work beyond the correct answer. Find it, click it and move on.