Problematics: Dates with Parliament

It can happen within the Council of Ministers — Ram Vilas Paswan’s 1946, Nitin Gadkari’s 1957 and Jayant Sinha’ 1963 were another set of years with a common calendar. It can happen among the same party’s MPs, such as the Shiv Sena’s Gajanand Kirtikar (1943), Vinayak Raut (1954) and Krupal Tumane (1965).

Why must a calendar repeat itself from time to time? Because there are only 14 possible calendars to recycle. Why 14? I had thought of putting that as a puzzle but then realised a Google search gives the game away. So here’s why.

If you make a calendar with New Year’s Day as a Sunday, say, then January 2 must be a Monday, the 3rd must be a Tuesday, and so on — all other dates automatically fall into place. You can begin a calendar with only 7 choices for January 1 — Sun, Mon, Tue… Sat. That’s 7 possible calendars for non-leap years. Add 7 more for leap years.

Is there a fixed cycle after which a series of calendars must repeat the same sequence? A Google search may lead you to a cycle of 28 years. Let’s test that model with my own example: 1952-80.

The calendar for 1952 (Sushma Swaraj) indeed returns 28 years later, in 1980 (Poonam Mahajan). Next comes D.V. Sadananda Gowda’s 1953, the calendar for which resurfaces after 28 years, in Anupriya Patel’s 1981. It works all the way down to 1979 (Dharmendra Yadav) corresponding with 2007 (the future of Parliament).

The 28-year model survived the 1952-80 test. It will, however, fail in certain sets of years. Meaning, you can still hope for an original puzzle.

Hold your patience. We will return to calendars after this very refreshing break.

What you wrote

Of the two problems last week, the responses to Puzzle#1B first. Here are three people with one view:

I am so glad that you started something like this. There is no reason to add the 2K and 27K — the 2K is contained within the 27K already.
Tushar Gupta, DoMS, IIT Madras
***
The three women paid a total of Rs 27,000. Of which Rs 25,000 is with the father and the remaining Rs 2,000 is with the son.
Nitin Kumbhar

***
The total money spent by the women = 27,000, which includes 25,000 rent and 2,000 for the boys’ pocket. So, it does not make sense to add 27,000 and 2,000. Keep writing. Cheers.
Sampath Kumar V, IIM-K alumnus

Again, here are two others in full agreement on another view:

You know what. Try counting both your hands’ fingers. First count your right-hand fingers but start from 10. I’m pretty sure the last digit, you will say, is 6 in your right hand. Now count your left-hand fingers but start from 1. Again I’m quite sure the last digit, you will say, is 5 in your left hand. Which means, 6 + 5 = 11. Something is wrong, right?
Anil Kumar
***
What we did was akin to the 10-finger count, holding up both hands, counting the fingers on one hand down from 10 to 6 and then adding 5 of the other hand to arrive at 11 fingers.
Aakar Patel

“You tricked us.” alleges Shalesh Kumar, as he and Natarajan separately break down the Rs 30,000 into its correct components. Abhinav Dutta notes that we got 29,000 “because we are deducting 3,000 and adding 2,000”.

Meanwhile, here’s the answer to Puzzle#1A:

Hi Kabir,
India defeated Ireland and Pakistan but lost to Zimbabwe and UAE.
Pakistan defeated Ireland and UAE but lost to India and Zimbabwe.
Ireland defeated Zimbabwe and UAE but lost to Pakistan and India.
UAE defeated Zimbabwe and India but lost to Pakistan and Ireland.
Zimbabwe defeated India and Pakistan but lost to Ireland and UAE.
Good job. Thank you.
Shalesh Kumar

Identical answers from Natarajan, Tushar Gupta and Sampath Kumar V.

The leap year fix

An unfortunately large number of people spend their entire lives under the notion that a leap year happens once every four years. A leap year comes once in approximately four years. Its average cycle is 4.1237 years, for it sometimes takes a break.

(a) Years that are multiples of 4 are leap years (eg. 1980, 2008) except that…
(b) Years ending in “00” are not leap years (eg. 1900, 2100) except that…
(c) “00” years beginning with a multiple of 4 are leap years (eg. 1600, 2000).

Because 2000 remained a leap year, the 28-year calendar cycle moved smoothly. So, year 2007 honoured the appointment with Dharmendra Yadav’s 1979.

But when 1900 converted and became a non-leap year, it upset the rhythm of the cycle. And year 1907 rejected the offer of partnership from year 1879, despite being 28 years apart.

Puzzle#2A: There is a very large cycle that all calendars must follow even after you factor in the leap year’s fickleness. If you take any year Y, its calendar must return in year Y + x. Find x.

Close of play

Puzzle#2B: Find a cricketing word of six letters from the following clues:
(a) The first letter is in SHAMI but not in MOHIT.
(b) Second letter is part of EXTRA-COVER but not of COVER.
(c) Third letter in UAE but not in SCOTLAND.
(d) Fourth letter with McCULLUM but not with CLARKE.
(e) Fifth letter in PULL but not in LOFT.
(f) Sixth letter enters BANGLADESH but not ENGLAND.

Please mail your replies to kabir.firaque@expressindia.com.