Henry Ernest Dudeney/Puzzles and Curious Problems/171 - Multiplication Dates/Solution

by : $171$

 * Multiplication Dates
 * In the year $1928$ there were $4$ dates which, when written in the form ,
 * the day multiplied by the month equal the year.
 * These are,  ,   and  .


 * How many times in the $20$th century -- inclusive -- does this so happen?


 * Or, you can try to find out which year in the century gives the largest number of dates that comply with the conditions.
 * There is one year that beats all the others.

Solution
According to, there are $215$ dates between $1$st January $1901$ and $31$st December $2000$ in such a form.

includes such dates as $25/4/00$, where he interprets $00$ as $100$.


 * $1924$ has the most such dates:


 * $24/1/24$
 * $12/2/24$
 * $8/3/24$
 * $6/4/24$
 * $4/6/24$
 * $3/8/24$
 * $2/12/24$

Of course, this applies to any century, not just the $20$th.

Proof
January:
 * $1/1/01$

to:
 * $31/1/31$

a total of $31$

February:
 * $1/2/02$

to:
 * $28/2/56$

a total of $28$

March:
 * $1/3/03$

to:
 * $31/3/93$

a total of $31$

April:
 * $1/4/04$

to:
 * $25/4/00$

a total of $25$

May:
 * $1/5/05$

to:
 * $20/5/00$

a total of $20$

June:
 * $1/6/06$

to:
 * $16/6/96$

a total of $16$

July:
 * $1/7/07$

to:
 * $14/7/98$

a total of $14$

August:
 * $1/8/08$

to:
 * $12/9/96$

a total of $12$

September:
 * $1/9/09$

to:
 * $11/9/99$

a total of $11$

October:
 * $1/10/10$

to:
 * $10/10/00$

a total of $10$

November:
 * $1/11/11$

to:
 * $9/11/99$

a total of $9$

Decmber:
 * $1/12/12$

to:
 * $8/12/96$

a total of $8$

Hence there are:
 * $31 + 28 + 31 + 25 + 20 + 16 + 14 + 12 + 11 + 10 + 9 + 8 = 215$

These can be enumerated:
 * $1/1/01$


 * $2/1/02$
 * $1/2/02$


 * $1/3/03$
 * $3/1/03$


 * $4/1/04$
 * $2/2/04$
 * $1/4/04$


 * $5/1/05$
 * $1/5/05$


 * $6/1/06$
 * $3/2/06$
 * $2/3/06$
 * $1/6/06$


 * $7/1/07$
 * $1/7/07$


 * $8/1/08$
 * $4/2/08$
 * $2/4/08$
 * $1/8/08$


 * $9/1/09$
 * $3/3/09$
 * $1/9/09$


 * $10/1/10$
 * $5/2/10$
 * $2/5/10$
 * $1/10/10$


 * $11/1/11$
 * $1/11/11$


 * $12/1/12$
 * $6/2/12$
 * $3/4/12$
 * $4/3/12$
 * $2/6/12$
 * $1/12/12$


 * $13/1/13$


 * $14/1/14$
 * $7/2/14$
 * $2/7/14$


 * $15/1/15$
 * $5/3/15$
 * $3/5/15$


 * $16/1/16$
 * $8/2/16$
 * $4/4/16$
 * $2/8/16$


 * $17/1/17$


 * $18/1/18$
 * $9/2/18$
 * $6/3/18$
 * $3/6/18$
 * $2/9/18$


 * $19/1/19$


 * $20/1/20$
 * $10/2/20$
 * $5/4/20$
 * $4/5/20$
 * $2/10/20$


 * $21/1/21$
 * $7/3/21$
 * $3/7/21$


 * $22/1/22$
 * $11/2/22$
 * $2/11/22$


 * $23/1/23$


 * $24/1/24$
 * $12/2/24$
 * $8/3/24$
 * $6/4/24$
 * $4/6/24$
 * $3/8/24$
 * $2/12/24$


 * $25/1/25$
 * $5/5/25$


 * $26/1/26$
 * $13/2/26$


 * $27/1/27$
 * $9/3/27$
 * $3/9/27$


 * $28/1/28$
 * $14/2/28$
 * $7/4/28$
 * $4/7/28$


 * $29/1/29$


 * $30/1/30$
 * $15/2/30$
 * $10/3/30$
 * $6/5/30$
 * $5/6/30$
 * $3/10/30$


 * $31/1/31$


 * $16/2/32$
 * $8/4/32$
 * $4/8/32$


 * $11/3/33$
 * $3/1/33$


 * $17/2/34$


 * $7/5/35$
 * $5/7/35$


 * $18/2/36$
 * $12/3/36$
 * $9/4/36$
 * $6/6/36$
 * $4/9/36$
 * $3/12/36$


 * $19/2/38$


 * $13/3/39$


 * $20/2/40$
 * $10/4/40$
 * $8/5/40$
 * $5/8/40$
 * $4/10/40$


 * $21/2/42$
 * $14/3/42$
 * $7/6/42$
 * $6/7/42$


 * $11/4/44$
 * $4/11/44$


 * $15/3/45$
 * $9/5/45$
 * $5/9/45$


 * $23/2/46$


 * $24/2/48$
 * $16/3/48$
 * $12/4/48$
 * $8/6/48$
 * $6/8/48$
 * $4/12/48$


 * $7/7/49$


 * $25/2/50$
 * $10/5/50$
 * $5/10/50$


 * $17/3/51$


 * $26/2/52$
 * $13/4/52$


 * $27/2/54$
 * $18/3/54$
 * $9/6/54$
 * $6/9/54$


 * $11/5/55$
 * $5/1/55$


 * $28/2/56$
 * $14/4/56$
 * $8/7/56$
 * $7/8/56$


 * $19/3/57$


 * $20/3/60$
 * $15/4/60$
 * $12/5/60$
 * $10/6/60$
 * $6/10/60$
 * $5/12/60$


 * $21/3/63$
 * $9/7/63$
 * $7/9/63$


 * $16/4/64$
 * $8/8/64$


 * $13/5/65$


 * $11/6/66$
 * $6/11/66$


 * $17/4/68$


 * $23/3/69$


 * $14/5/70$
 * $10/7/70$
 * $7/10/70$


 * $24/3/72$
 * $18/4/72$
 * $12/6/72$
 * $9/8/72$
 * $8/9/72$
 * $6/12/72$


 * $25/3/75$
 * $15/5/75$


 * $19/4/76$


 * $11/7/77$
 * $7/11/77$


 * $26/3/78$
 * $13/6/78$


 * $20/4/80$
 * $16/5/80$
 * $10/8/80$
 * $8/10/80$


 * $27/3/81$
 * $9/9/81$


 * $28/3/84$
 * $21/4/84$
 * $14/6/84$
 * $12/7/84$


 * $17/5/85$


 * $1/1/86$


 * $29/3/87$


 * $1/1/88$


 * $1/1/89$


 * $30/3/90$
 * $18/5/90$
 * $15/6/90$
 * $10/9/90$
 * $9/10/90$


 * $13/7/91$


 * $23/4/92$


 * $31/3/93$


 * $19/5/95$


 * $24/4/96$
 * $16/6/96$
 * $12/8/96$
 * $8/12/96$


 * $14/7/98$


 * $11/9/99$
 * $9/11/99$


 * $25/4/00$
 * $20/5/00$
 * $10/10/00$

Of all these years, it is seen that $24$ has $7$ such dates, as enumerated.