Ultra-Quick Recall (What you’ll use in this chapter)

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1. What is parity?
   Whether a number is even or odd.
2. State \( \text{odd}+\text{odd}\) parity.
   Even.
3. State \( \text{even}+\text{odd}\) parity.
   Odd.
4. State \( \text{even}-\text{even}\) parity.
   Even.
5. Largest possible “taller-in-front” number in a line of 8 children?
   \(7\) (for the last child if all 7 in front are taller).
6. If someone is the tallest, what number do they call (counting taller-in-front)?
   \(0\) (no one in front is taller).
7. Sum of small squares in a \(27\times13\) grid: odd or even?
   Odd (odd\(\times\)odd).
8. Sum of small squares in a \(42\times78\) grid: odd or even?
   Even (even factor present).
9. 100th odd number?
   \(2\times100-1=199\).
10. Expression for the \(n\)th even number.
    \(2n\).
11. Expression for the \(n\)th odd number.
    \(2n-1\).
12. Magic sum of any \(3\times3\) magic square using \(1\)–\(9\)?
    \(15\).
13. Centre of that magic square (using \(1\)–\(9\))?
    \(5\).
14. Next three terms after \(55\) in \(1,2,3,5,8,\dots\)
    \(89,\,144,\,233\).
15. How many 8-beat rhythms with 1-beat & 2-beat syllables?
    \(34\).
16. Toggle a lamp 77 times (starts ON). Final state?
    OFF (odd toggles flip).
17. Solve: \(T+T+T=UT\). Find \(T,U\).
    \(T=5,\,U=1\) since \(3T=10U+T\Rightarrow2T=10U\).
18. Solve: \(K2+K2=HMM\). Find \(H,M,K\).
    \(H=1,\,M=4,\,K=7\) (since \(72+72=144\)).
19. Does \(4m-1\) always give an odd number?
    Yes.
20. Parity of the sum \(1+2+\cdots+100\)?
    Even (sum \(=5050\)).

1. Show that the sum of any 5 odd numbers is odd or even?
   Odd: \( \underbrace{\text{odd}+\text{odd}}_{\text{even}}+\underbrace{\text{odd}+\text{odd}}_{\text{even}}+\text{odd}=\text{even}+\text{odd}=\text{odd}.\)
2. Explain why 5 odd number cards cannot sum to \(30\).
   Sum of 5 odds is odd; \(30\) is even. Impossible.
3. Maria & Martin are consecutive ages. Can their sum be \(112\)?
   No. \(n+(n+1)=2n+1\) is odd; \(112\) is even.
4. Parity of small squares in a \(135\times654\) grid?
   Even (factor \(654\) is even).
5. Find a formula that always outputs even numbers.
   \(2k,\; 6p+8,\; 4n-2\) etc. (all multiples of \(2\)).
6. Find a formula that always outputs odd numbers.
   \(2n+1,\; 4q-1,\; 6t+5\) etc.
7. Which terms in \(1,2,3,5,8,\dots\) are even?
   Every 3rd term starting from \(2\): \(2,8,34,144,\dots\)
8. Write the general \(3\times3\) magic square with centre \(m\) and its magic sum.
   \(\begin{matrix}m+3&m-4&m+1\\ m-2&m&m+2\\ m-1&m+4&m-3\end{matrix}\), sum \(=3m\).
9. If centre \(m=20\), build the magic square and give the magic sum.
   \(\begin{matrix}23&16&21\\ 18&20&22\\ 19&24&17\end{matrix}\), magic sum \(=60\).
10. Explain why centre of a \(1\)–\(9\) magic square must be \(5\).
    Magic sum \(=15\). Centre is used in 4 lines; parity/feasibility checks show \(1\) or \(9\) at centre makes a \(15\) line impossible; only \(5\) works with remaining pairs to make \(15\).
11. Show that the total of the three row sums equals \(45\).
    Each number \(1\)–\(9\) appears in exactly one row. So total \(=\sum_{k=1}^9 k=45.\)
12. State whether: “First person’s taller-in-front number is 0.”
    Always true (no one in front).
13. State whether: “Person saying 0 must be tallest.”
    Only sometimes true (they could just be tallest among those in front).
14. Solve the alphametic \(YY+Z=ZOO\) with distinct digits.
    \(99+1=100\Rightarrow Y=9, Z=1, O=0\).
15. Solve the alphametic \(C1+C=1FF\).
    Units: \(1+C=10\Rightarrow C=9\) with carry \(1\). Tens: \(9+1=10\Rightarrow F=0\). So \(91+9=100\Rightarrow C=9,F=0.\)
16. Find parity of the sum of 8 odd numbers and 3 even numbers.
    \(\underbrace{\text{odd}+\cdots+\text{odd}}_{8=\text{even count}}\Rightarrow \text{even}\); even + even + even \(=\) even. Final: even.
17. Find parity of \( (3n+4)\) for \(n=3\) and \(n=8\).
    \(n=3\Rightarrow13\) (odd), \(n=8\Rightarrow28\) (even). It can be either.
18. Compute the next two Virahāṅka numbers after \(987,1597\).
    \(2584,\,4181\) (since \(987+1597=2584,\;1597+2584=4181\)).
19. How many 6-beat rhythms?
    \(13\) (because \(F(6)=F(5)+F(4)=8+5\)).
20. Solve \(UT+TA=TAT\).
    \(A=0,\;U+T=10,\;T=1,\;U=9\Rightarrow 91+10=101.\)

1. Prove that the sum of any even number of odd numbers is even.
   Pair the odds: \((2a+1)+(2b+1)=2(a+b+1)\) (even). With an even number of odds we can fully pair them, so total is even.
2. Show that the product of an odd and an even number is even.
   \((2k+1)\cdot (2m)=2[(2k+1)m]\) which is a multiple of \(2\), i.e., even.
3. Explain why a \(3\times3\) grid with a row sum \(5\) or \(26\) is impossible when using \(1\)–\(9\).
   Smallest row sum \(=1+2+3=6\). Largest \(=7+8+9=24\). So \(5\) or \(26\) are impossible.
4. Using the general magic square (centre \(m\)), find the square when \(m=25\). State the magic sum.
   \(\begin{matrix}28&21&26\\ 23&25&27\\ 24&29&22\end{matrix}\), magic sum \(=75\).
5. Create a 3×3 magic square with magic sum \(=60\).
   Take \(m=20\): \(\begin{matrix}23&16&21\\ 18&20&22\\ 19&24&17\end{matrix}\). Check: each line \(=60\).
6. Show that adding a constant \(c\) to each entry in a magic square increases the magic sum by \(3c\).
   Each line has 3 entries, so line sum \(\to\) old sum \(+3c\).\
7. Show that doubling every entry in a magic square doubles the magic sum.
   Every line’s sum is multiplied by \(2\); hence magic sum doubles.
8. Find the parity of the 20th Virahāṅka number.
   Pattern O,E,O,O,E repeats every 3. Since \(20\equiv 2\pmod3\), it is even.
9. Number of ways to climb 8 steps taking 1 or 2 at a time.
   Same recurrence as beats: \(F(8)=34\).
10. Prove that \(2n-1\) lists all odd numbers and \(2n\) lists all even numbers (for \(n\in\mathbb{N}\)).
    Any integer is either \(2q\) (even) or \(2q+1\) (odd). As \(n\) varies, \(2n\) hits all evens; \(2n-1\) hits all odds.
11. From \(YY+Z=ZOO\), derive \(Y=9Z+O\) and conclude the digits.
    \(11Y+Z=100Z+11O\Rightarrow Y=9Z+O\). Only \(Z=1,O=0,Y=9\) fits single-digit constraint.
12. Show that the sum of three consecutive integers is a multiple of 3.
    Let them be \(n-1,n,n+1\). Sum \(=3n\), divisible by \(3\).
13. If \(n\) is even, show \(n^2\) is even; if \(n\) is odd, show \(n^2\) is odd.
    Even: \(n=2k\Rightarrow n^2=4k^2\) even. Odd: \(n=2k+1\Rightarrow n^2=4k(k+1)+1\) odd.
14. For centre \(m\), prove each row/column sum in the general square is \(3m\).
    Top row: \((m+3)+(m-4)+(m+1)=3m\). Similar for others; diagonals also \(=3m\).
15. Give a reason why \(1\) and \(9\) cannot sit in any corner in the \(1\)–\(9\) magic square.
    A corner participates in two lines with centre \(5\). The remaining needed numbers to make \(15\) cannot be simultaneously placed without conflict if the corner is \(1\) or \(9\). Hence they must be on edges.
16. Is it possible to make a magic square with nine non-consecutive numbers?
    Yes. Multiply any magic square by a constant \(k>1\) (and/or add a constant). The pattern remains magic; entries aren’t consecutive.
17. Show that sum of row sums equals sum of column sums in a filled \(3\times3\) grid.
    Both equal the total of all 9 entries (each counted once).
18. Explain why \(6k+2\) lists only some even numbers.
    It hits numbers \(\equiv2\pmod6\) (e.g., \(2,8,14,\dots\)), not all evens like \(4,10,\dots\).
19. State the parity of \( (2a+1)+(2b+1)+(2c+1)+(2d+1)\).
    Even \((=2(a+b+c+d+2))\).
20. For the height rule, classify: “The person with the largest called number is the shortest.”
    Only sometimes true (largest count just means many taller in front; someone behind may still be shorter).

6.1 Numbers Tell us Things (Taller-in-Front Rule)

E1. For each statement, mark Always True, Only Sometimes True, or Never True:

1. (a) If a person says ‘0’, then they are the tallest in the group.
2. (b) If a person is the tallest, then their number is ‘0’.
3. (c) The first person’s number is ‘0’.
4. (d) Someone in the middle cannot say ‘0’.
5. (e) The person who calls out the largest number is the shortest.
6. (f) Largest possible number in a group of 8 people?

(a) Only sometimes true. (b) Always true. (c) Always true. (d) Never true (they can be tallest among those in front).
(e) Only sometimes true. (f) \(7\).

E2. Construct any lineup for the sequence \(0,1,1,2,4,1,5\) (heights ranked 7 = tallest to 1 = shortest).

One valid construction exists (build left to right ensuring each person’s count equals “taller in front”). Example order by height ranks:
\((6,\,7,\,5,\,3,\,1,\,4,\,2)\). (Check: counts in front taller \(=\) \(0,1,1,2,4,1,5\).)

Tip: Build left→right. For person \(i\), ensure exactly the required number of taller people appear before position \(i\).

6.2 Picking Parity

E3. Find parity:

1. (a) even + even + odd + odd
2. (b) odd + odd + even + even + even
3. (c) sum of 5 even numbers
4. (d) sum of 8 odd numbers

(a) even; (b) even; (c) even; (d) even.

E4. Lakpa has an odd # of ₹1 coins, odd # of ₹5 coins, even # of ₹10 coins. Total shown ₹205. Did he err?

Yes. Total parity \(=\) odd (₹1) \(+\) odd (₹5) \(+\) even (₹10) \(=\) even, but \(205\) is odd. Contradiction.

E5. Complete parity for subtraction:

(d) even − even = even; (e) odd − odd = even; (f) even − odd = odd; (g) odd − even = odd.

E6. Parity of small squares in grids:

(a) \(27\times13\Rightarrow\) odd; (b) \(42\times78\Rightarrow\) even; (c) \(135\times654\Rightarrow\) even.

E7. Parity of expressions and nth terms:

Always even: \(2k,\;6p+8\). Always odd: \(2n+1,\;4q-1\). Mixed: \(3n+4\).
\(n\)th even: \(2n\). \(n\)th odd: \(2n-1\). 100th odd \(=199\).

6.3 Some Explorations in Grids

E8. Why is the grid with circle sums \(5\) or \(26\) impossible (using digits \(1\)–\(9\) once)?

Row/column sum min \(=1+2+3=6\), max \(=7+8+9=24\). So \(5,26\) are impossible.

E9. Show that sum of all three row sums equals \(45\); same for columns.

Every number \(1\)–\(9\) is counted exactly once across rows, so total \(=1+2+\cdots+9=45\). Similarly for columns.

E10. Magic square basics with \(1\)–\(9\): (i) Magic sum? (ii) Centre? (iii) Build one.

(i) \(15\). (ii) \(5\). (iii) One classic: \(\begin{matrix}8&1&6\\ 3&5&7\\ 4&9&2\end{matrix}\).

E11. Generalise a 3×3 magic square with centre \(m\). Then build one for \(m=25\).

General: \(\begin{matrix}m+3&m-4&m+1\\ m-2&m&m+2\\ m-1&m+4&m-3\end{matrix}\). For \(m=25\): \(\begin{matrix}28&21&26\\ 23&25&27\\ 24&29&22\end{matrix}\).

E12. If each entry is increased by \(1\) (or doubled), what happens to the magic sum?

Add \(1\): sum increases by \(3\). Double: sum doubles.

6.4 Virahāṅka–Fibonacci Numbers

E13. Count 5-beat rhythms (1-beat & 2-beat). Verify with recurrence.

\(F(5)=F(4)+F(3)=5+3=8\). (Also list via 1+ of 4-beats and 2+ of 3-beats.)

E14. How many 6-beat rhythms? How many 8-beat rhythms?

\(F(6)=13,\; F(8)=34.\)

E15. Next 3 terms after \(55\) in \(1,2,3,5,8,\dots\)

\(89,\,144,\,233\).

E16. Parity pattern of the sequence?

Repeats every 3 terms: odd, even, odd, odd, even, … (every 3rd term is even).

6.5 Digits in Disguise (Cryptarithms)

E17. Solve \(T+T+T=UT\).

\(T=5, U=1\) (\(3T=10U+T\Rightarrow 2T=10U\)).

E18. Solve \(K2+K2=HMM\).

\(K=7, M=4, H=1\) since \(72+72=144\).

E19. Solve \(YY+Z=ZOO\) (distinct letters).

\(99+1=100\Rightarrow Y=9, Z=1, O=0.\)

E20. Solve \(C1+C=1FF\).

\(91+9=100\Rightarrow C=9, F=0.\)

E21. Toggle puzzle: bulb starts ON; switch toggled \(77\) times. Final state?

OFF (odd toggles flip state).

E22. Parity grid (2×3 with row/col parity e/o). Fill with 3 odds & 3 evens.

One fill: first row \(o,e,o\) (row sum \(=e\)); second row \(e,o,e\) (row sum \(=o\)). Columns become \(e,e,o\) as required. (Any matching pattern is fine.)

E23. 3×3 magic square with magic sum \(0\) (use negatives; not all zeros).

Take \(m=0\): \(\begin{matrix}3&-4&1\\ -2&0&2\\ -1&4&-3\end{matrix}\) has sum \(0\) in every row/col/diag.

E24. Fill: (a) sum of an odd number of evens; (b) sum of an even number of odds; (c) sum of an even number of evens; (d) sum of an odd number of odds.

(a) even; (b) even; (c) even; (d) odd.

E25. Parity of the sum \(1+2+\cdots+100\).

Even (\(=5050\)).

E26. Two consecutive Virahāṅka numbers are \(987,1597\). Next two? Previous two?

Next: \(2584,4181\). Previous: \(610,377\) (backwards: \(1597-987=610,\; 987-610=377\)).

E27. Angaan climbs 8 steps (1 or 2 at a time). In how many ways?

\(34\) ways.

E28. Parity of the 20th Virahāṅka number.

Even (position \(20\equiv 2\pmod3\)).

E29. True/False: (a) \(4m-1\) always odd; (b) all evens are \(6j-4\); (c) all odds are \(2p+1\) and also \(2q-1\); (d) \(2f+3\) gives both parities.

(a) True; (b) False (misses numbers \(\equiv 0,2 \pmod6\)); (c) True; (d) True.

E30. Cryptarithm \(UT+TA=TAT\). Solve.

\(U=9, T=1, A=0\Rightarrow 91+10=101\).

Neat Examples (MathJax formatting)