Deduction and clues: open practice pack: worked explanations

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This is a rule applied to a case: the step logicians call modus ponens. One sentence is a general rule covering everything on the trolley; the other says the fern is one of those things, so the rule attaches to it and no further checking is needed. The tempting wrong route reads the rule backwards, as though only trolley plants need daily water; nothing said that, and a fern on a windowsill could need water just as often. Whenever a sentence opens with 'every', look for a case that satisfies its opening condition and read the conclusion straight off.

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This is the contrapositive: attending implies a collected badge, so no badge implies no attending. Flipping a rule and negating both halves preserves its truth exactly. Test it directly. Suppose she had attended; the rule would then hand her a badge, contradicting what you were told, so attendance is ruled out. The near-miss keeps her at the briefing and treats the missing badge as an oversight, but the rule as stated leaves no such gap. Note also that the rule says nothing about people who did collect a badge, so it never tells you what badge holders did.

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This is chaining by transitivity. Both comparisons mention the red parcel, so they link at that point: green sits above red, red sits above grey, and the single chain gives the whole order without weighing anything. The most common slip is reading the whole chain the wrong way round and reporting it lightest first, which reverses the line completely. The other trap is keeping the ends right but swapping the middle, which quietly contradicts the comparison that put red above grey. Look for the item that appears in two clues; that shared item is always the hinge.

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A universal rule lets you relabel. Take the loaves that were sold before noon: they were baked this morning, and everything baked this morning is wholemeal, so those same loaves are wholemeal. A 'some' claim carried across an 'all' claim stays a 'some' claim. The near-miss inflates the quantifier and reports that every wholemeal loaf sold early, which the premises never support: yesterday's wholemeal stock could still be sitting on the shelf. Watch the quantifier you started with and refuse to let it grow while it travels.

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A 'no A does B' statement says two groups never overlap: lorries and things entering before seven have nothing in common. The vehicle is squarely in the second group, so it cannot be in the first, and that is the whole step. One tempting reading treats the sentence as a permission that someone might have flouted, but it describes what happens at the depot, not what would be allowed. Another names a specific replacement, and nothing here narrows the field to vans: a car, a bicycle or a milk float would fit just as well. Take a 'no' statement as an exclusion, and claim only the exclusion.

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Denying the antecedent is the classic invalid step. The rule promises what happens when the alarm sounds and says nothing whatever about a quiet morning, so a silent alarm licenses no conclusion about the doors. A caretaker, a timer or a delivery driver could have locked them for reasons the rule never mentions. The near-miss slides from 'sounding locks the doors' to 'only sounding locks the doors', quietly adding an exclusivity the sentence never claimed. When you are handed the negation of an if-clause, expect nothing to follow and check whether the rule was ever stated the other way round.

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This is affirming the consequent. A conditional guarantees the effect once the cause is present; it never promises that the cause is the only route to that effect. Rain, a burst hose or a tipped watering can all leave the same wet path, and the rule excludes none of them, so observing the effect leaves the sprinkler an open question. The confident wrong route runs the arrow backwards and reports that the sprinkler must have run. Read every 'if … then' as a one-way arrow, and check which end of it your evidence sits at before drawing anything.

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Turn each comparison into a link and join them at their shared names. The first two clues give Jen above Kofi above Ivo; the third clue hooks on at the top, putting Lena above Jen, so the full line runs Lena, Jen, Kofi, Ivo and the winner is the name at the head of it. Stopping after the first two clues leaves Jen apparently in front: the commonest error here, and the reason to check that every clue has been used before reading anything off. Reading all three relations the wrong way round produces the exact reverse of the line, with the runner who actually came last on top.

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Start with the claim that undermines itself. If the speaker who says they are both truthful were the truthful one, his claim would be true and both would be honest, which contradicts the stated fact that only one of them is. So that claim is false, its speaker is the liar, and the other speaker can be trusted completely. Her two facts then combine directly: the meeting was Monday, and the delivery came the day after it. Reading 'the day after' as the day before lands you a day early; counting on past the following day lands you a day late. Whenever a speaker's claim would break the setup if true, that is the thread to pull first.

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Two universal statements chain when the middle group is shared. Everything in the tray sits inside the marigold group, and the marigold group sits entirely outside the treated group, so nothing in the tray can be treated. An 'all' followed by a 'no' yields a 'no'. Both wrong routes reverse a statement instead of chaining it: one turns 'no marigold is treated' into a claim about treated plants being marigolds, the other turns 'all tray-one plants are marigolds' into a claim that untreated plants live in the tray. Draw the groups as nested and separated regions and the direction of each claim stays visible.

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The decisive move is combining a weak comparison with a clue that forbids equality. 'At least as many' allows two readings (more, or exactly the same), and on its own it settles nothing. The no-ties clue removes the second reading, so it becomes strictly more, and the chain closes: Ash above Brook above Dawn above Clay. The near-miss treats the weak comparison as an equality and leaves the bottom two tied or the wrong way round. Any time a clue uses 'at least', 'no more than' or 'not later than', look for a second clue that rules the boundary case in or out.

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Do not argue about who sounds honest; test each candidate and count. If the accused colleague forgot, his own denial fails and so does the accusation aimed at someone else: two false statements, one too many. If either of the two remaining colleagues forgot, both accusations fail, again two. The count only comes out right for the colleague who is both an accuser and the person accused in the last statement: her own accusation is false, the denial is true and the accusation against her is true, giving exactly one falsehood. The trap is treating two statements pointing away from her as weight of evidence; a counting constraint is arithmetic, not a vote.

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Place the block first. 'Immediately before' welds two workshops into a two-morning unit that can only sit at the start or the end of the week, and the other workshop takes whichever morning is left. Putting the unit at the end forces first aid onto Monday, which one clue forbids, so the unit sits at the start and first aid closes the week. Two familiar errors show up here: loosening 'immediately before' into 'somewhere before', which lets a third workshop slide between the pair, and reversing the pair so that the later workshop is placed first. With adjacency clues, always try the block in each legal position before assigning anything else.

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The flaw is the leap from sequence to cause, after this, therefore because of this. The facts fix an order of events and nothing more: a kerbed wheel, an ageing battery or a fortnight of hard driving would produce the same April breakdowns with a perfectly good March service behind them. Until something rules those out, the service is one candidate cause among several. The tempting objection about the small number of breakdowns misses the real gap, because even a hundred failures would still leave the cause unlocated; what is missing is not volume of evidence but any evidence that separates the candidates. Ask what else could have produced the observation before naming a cause.

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Nested conditionals are peeled from the outside in. The outer rule fires because its condition, a cancelled ferry, is given, and what it delivers is the inner rule: open bridge means bridge route. You are told the bridge route was not taken, so running that inner rule backwards leaves only one possibility for the bridge. Two wrong routes are worth naming: one rejects a fact you were handed rather than a fact you derived, and one turns a single day's outcome into a standing rule about every cancellation. Peel the outer layer with the fact you are given, then apply the flipped form to the layer underneath.

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Negative clues work by stacking. One person is barred from bread and cheese by one clue and from the tart by another, which leaves exactly one item for him, and that single forced assignment is what unlocks everything else. With the cheese already named and the olives now taken, the neighbour barred from olives and tart can only have brought the bread, and the last name takes what remains. The near-miss is to fix only the directly stated pairing and then guess among the rest; the clue that never names an item positively is doing the heavy work. Count exclusions per person and act on whoever has the fewest items left.

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A majority claim survives being pushed through an exception-free rule, but it never grows on the way. Picture ten members with seven owning waterproofs: all seven have walked the path, so a majority have walked it and certainly at least one has, yet the three without waterproofs are untouched by the rule and may never have been near the coast. That single model is enough to show which conclusion overreaches. The mistake to watch for is hearing 'most' as 'effectively everyone' once a firm second rule appears; the firmness belongs to the rule about owners, not to the size of the group it covers.

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Run the count for each candidate rather than weighing accusations. Take the doubly accused colleague first: if he booked the room, both accusations against him hold and so does the lone denial, leaving only one false statement. Short of the required two. If the denier booked it, every statement fails, giving four. If the fourth colleague booked it, three fail. Only the person who accused the doubly accused colleague produces exactly two falsehoods: her own accusation and the matching one beside it. The trap is the pile-on. Two people naming the same colleague feels decisive, when in fact it is precisely why that case cannot supply enough falsehoods.

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Fix the absolute clue, then slide the adjacent pair. With the back of the queue taken, the welded pair can start at position one, two or three. Starting at one puts the second member of the pair in second place, which a clue forbids. Starting at two leaves places one and four for the remaining two people, and the only way to fill them puts the wrong person in fourth. Starting at three works: the pair takes third and fourth, and the ahead-of clue orders the front two, giving the whole line. Dropping the clue about fourth place is what produces the tempting alternative in which the pair sits earlier; counting the queue from the back instead of the front is the other reliable way to go wrong.

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'Only if' marks a necessary condition, not a sufficient one. The rule points one way, entering implies having paid, so it tells you what every entrant must have done, and nothing about what a payer may now do. The club could also require membership, a place in the draw or a partner, and the rule as written leaves all of that open. Two errors sit close together here: running the arrow backwards so that payment unlocks entry, and quietly upgrading permission into fact by treating a paid fee as evidence she took part. Rewrite any 'only if' as an arrow from the permitted act to its requirement, then check which end your evidence sits at.

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Test redundancy by deletion: cover a clue and ask whether the remaining ones still force a single rota. Once Monday is claimed outright, the clue keeping one colleague off Monday adds nothing that the other clues have not already settled: it repeats information rather than removing a possibility. Everything else earns its place: the adjacent pair can sit in two positions across the week, and the clue barring Thursday is the decisive one that kills the earlier of those two, leaving Tuesday, Wednesday and Thursday filled in only one way. Redundant does not mean false, and a clue that merely feels obvious may still be the one doing the eliminating. The deletion test is what separates them.

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This is reasoning from what someone does not know. Work out what the speaker would have said in each case: if the counter she was looking at had been the blue one, every counter left in play would be red and she could have named her own immediately. She could not, so the counter facing her is not the blue one, and that fixes the colour of the counter its holder cannot see. Her own counter stays open, which is the near-miss to avoid: the inference travels from her ignorance to what she can see, not to what she holds. Treating 'I do not know' as empty is the other trap; an admission of ignorance eliminates every situation in which the speaker would have known.

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This is case analysis under a counting bound, and the discipline is to build the whole table before judging anything. Test each person as the culprit and mark all five statements: the accuser's own case yields three true statements, the denier's yields two, and the two colleagues nobody named yield three apiece. One case yields four. The pair-accusation holds, the denial holds, both endorsements hold, and only the claim that one speaker is lying fails. Since the bound demands at least four, exactly one case survives. Two things trip solvers here: assuming the culprit must also be a liar, which nothing states, and accepting a three-true case because it is internally consistent. Consistency is not the test, the stated count is.

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This is proof by contradiction. Suppose the runner tied to an either/or clue came first. The conditional then fires and places Ines ahead of Gwen, Gwen is already ahead of Hugo, and only one order is left, which puts Ines in second place, exactly what the final clue forbids. The assumption cannot stand, so the other branch of the either/or is the only one available and the last place is settled. Notice that the remaining three places are still open and you never needed them: aim an assumption at the fact you were actually asked about and stop the moment it is decided. Following the rejected branch to the end is what produces the tempting result that the runner beaten by Gwen came last.

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Before answering, try to build two worlds instead of one. Both of these satisfy every clue: Otto in week one, Mira in week two, Noor in week three, Pia in week four; and Otto in week one, Pia in week two, Noor in week three, Mira in week four. Mira sits in a different week in each, so nothing here pins her down. The clues are not useless (two colleagues land in the same week in both worlds, and one week is genuinely closed to her), so the honest verdict is that some facts follow and this one does not. The mistake is stopping at the first arrangement you manage to build and reading the answer off it: an arrangement that merely fits is not an arrangement that is forced.