Reasoning formulas

Always equivalent to the original. The converse $Q\to P$ and the inverse $\neg P\to \neg Q$ do NOT follow from $P\to Q$ (they are equivalent to each other, not to the original).

'If P, then Q' is false only when P is true and Q is false. So the denial of 'If it rains, the match is called off' is 'It rains and the match is not called off', not 'If it rains, the match is not called off'.

'Only if' introduces the necessary condition. 'Unless' means 'if not'. 'P if and only if Q' means both $P\to Q$ and $Q\to P$.

The negation of 'all' is 'some … not', not 'none'. Use these to find contradictory pairs and to run the negation test for assumptions.

Read 'or' as inclusive (one or both) unless the wording makes it exclusive ('but not both', 'exactly one of').

$n$-th term of an arithmetic series (constant difference $d$) and of a geometric series (constant ratio $r$). Positions are counted from $n=1$.

Applies when the second differences are all equal to $c$; ${\Delta }_1=a_2-a_1$ is the first difference. Example: 5, 11, 23, 41, 65 has ${\Delta }_1=6$ and $c=6$, so $a_6=5+5\times 6+10\times 6=95$.

Opposite (mirror) pairs A–Z, B–Y, …, M–N have positions that add to 27. A forward shift of $k$ with wrap-around gives position $((n+k-1)\mathrm{mod}26)+1$, so Y shifted forward by 3 is B. A backward shift of $k$ is the same as a forward shift of $26-k$, so C shifted back by 3 is Z.

For one person who is $L$-th from the left and $R$-th from the right in a single row of $N$ people. The same holds for ranks from the top and the bottom of a list with no ties.

X is $a$-th from the left and Y is $b$-th from the right in a row of $N$. If $a+b=N+1$, X and Y are the same person. Convert both to positions from the same end if in doubt: Y is at position $N-b+1$ from the left.

Ways to seat $n$ distinct people. Around a circle only relative positions matter, so rotations count as one arrangement. Divide by 2 only when clockwise and anticlockwise orders are not distinguished, as for a garland of distinct beads ($n\ge 3$).

Take east and north as positive, west and south as negative. Name the direction by its quadrant (for example south-east when the net movement is east and south), or by the axis if one component is zero.

At H hours M minutes (H on the 12-hour dial, using 0 for 12) the hour hand is $30H+0.5M$ degrees past 12 and the minute hand is $6M$ degrees past 12, both measured clockwise.

Angle in degrees at H hours M minutes, with H the hour on the 12-hour dial (use 0 for 12). Example: at 7:24, $\theta =|210-132|=78^{\circ }$.

Minutes past H o'clock (H from 0 to 11, using 0 for 12) at which the hands are $\theta$ degrees apart, for $0^{\circ }\le \theta \le 180^{\circ }$. These values solve $30H-\frac{11}{2}M=\pm \theta$ or $\pm (360^{\circ }-\theta )$. Keep only values with $0\le M<60$. In the first pair, the minus sign gives a moment before the minute hand passes the hour hand and the plus sign a moment after. Try the second pair whenever a value from the first pair falls outside the range: between 1 and 2 o'clock the hands are at right angles at $M=\frac{2}{11}(30+90)=21\frac{9}{11}$ and at $M=\frac{2}{11}(30+270)=54\frac{6}{11}$, and the second time comes only from the second pair.

For a correct clock. In 12 hours the hands coincide 11 times, are opposite 11 times, and are at right angles 22 times (any angle strictly between $0^{\circ }$ and $180^{\circ }$ occurs 22 times). In 24 hours each count doubles.

For an upright plane mirror beside the clock, which swaps left and right. H is the hour on the 12-hour dial (use 0 for 12), and a result of 0 hours is read as 12. It is the same as subtracting the actual time from 12:00: the image of 4:20 reads 7:40, the image of 3:00 reads 9:00 and the image of 12:20 reads 11:40.

2028 and 2000 are leap years; 2100 and 1900 are not.

The pattern repeats every 400 years, so years 1601 to 1900 also give 1 odd day. Months: 31 days = 3 odd days, 30 days = 2, February = 0 (1 in a leap year).

Count the complete years before the given year (using the values above), then the complete months of the given year, then the day of the month itself (Gregorian rules applied throughout).

The weekday of a date moves forward by 1 each year, and by 2 when a 29 February falls between the two dates. From Saturday 15 August 2026 to 15 August 2031 the shift is $5+1=6$ (29 February 2028 lies between), so 15 August 2031 is a Friday.

A definite relation follows only if every sign on the path between the two terms points the same way; an = sign fits either direction and can sit anywhere in the chain. The relation is = if every sign on the path is =, strict ($>$ or $<$) if at least one sign is strict, and otherwise $\ge$ or $\le$.

Signs that point in opposite directions (here B is smaller than both) leave A and C undetermined. The same holds for $A<B>C$ and for $A\ge B\le C$.

Multiplying or dividing both sides by a negative number reverses the sign. Squaring is safe only when both sides are non-negative: $-3<2$, but $9>4$.

$N$ is the total number of people and $|A|$ the number in group A. The number in A only is $|A|-|A\cap B|$.

Each pairwise count includes the people who are in all three groups.

Here $S_1=|A|+|B|+|C|$, $S_2=|A\cap B|+|B\cap C|+|C\cap A|$ and $S_3=|A\cap B\cap C|$. Both follow from $S_1=e_1+2e_2+3e_3$ and $S_2=e_2+3e_3$, where $e_k$ is the number of people in exactly $k$ of the groups.

For two groups drawn from the same $N$ people, when nothing else is known. If 70% of a class like tea and 60% like coffee, at least 30% and at most 60% of the class like both.