Calendar Capers

Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat this for a number of your choice from the second row. You should now have just one number left on the bottom row, circle it. Find the total for the three numbers circled. Compare this total with the number in the centre of the square. What do you find? Can you explain why this happens?

Hockey

After some matches were played, most of the information in the table containing the results of the games was accidentally deleted. What was the score in each match played?

Clock Squares

Square numbers can be represented on the seven-clock (representing these numbers modulo 7). This works like the days of the week.

Tree Tops

Stage: 3 Challenge Level:

There were several answers to this problem but they were not solutions or explanation of a strategy so I have not included them. There are many possible solutions to this problem, which requires some sense making as well as some calculating. I therefore liked the solution from Laura of Teesside High School very much indeed. It showed some analysis and some good common sense (a very useful talent for a mathematician to have). This is followed by the solution sent in by Andrei of School 205 Bucharest.

To maximise the profit from the crop of trees both the initial expenditure and the long-term profit must be considered.

The Lodgepole Ping is the most expensive and the European Larch the cheapest of the trees. Then this is offset against the profit gained from the thinnings at both 10 and 20 years:

Sitka Spruce: -120000+10000+40000= -70000
European Larch: -158000+15000+40000= -60000
Lodgepole Pine: -130000+20000+30000= -80000

So over the first 20 years the European Larch loses the least money. However, to discover the best tree to plant for a long term profit over 70 years this loss must be taken from the profit gained when the trees are felled:

SS: 1126800-70000= 1056800
EL: 1158000-60000= 1098000
LP: 1144000-80000= 1064000

So after 70 years the European Larch makes the most profit. However after 70 years two of the trees begin to be worth less so if the trees are to be felled at 90 years the profit must be recalculated.

SS: 805000-70000=735000
EL: 837000-60000=777000
LP: 1476000-80000=1396000

So after 90 years the Lodgepole Pine is the best value, surpassing the profit gained by the European Larch after 70 years, though not by a great deal.

So perhaps for a more long term plan it would be better to plant a mixture of both Lodgepole Pine and European Larch, to reduce the initial loss and to take some revenue from the Larch after 70 years, and then replant these fields with some of the profits, then taking a larger profit at 90 years. This would give a cycle so that some money would be collected after a shorter period of time to sustain the plantation, as the extra £298000 gained with the Lodgepole Pine must be compared with the twenty years extra for this gain.

***

I observed that in the table the possible income per hectare after a number of years was written. I created a table with the profit for each type of tree, taking into account the planting cost per hectare, the profit from the thinning after 10 and 20 years (per hectare too), and the possible income after the period of time written above.

 After Years Profit (£) Sitka Spruce European Larch Lodgepole Pine 30 288000 132000 42500 40 443000 409200 286400 50 623000 798000 566000 60 764000 1124000 870200 70 1056800 1098000 1064000 80 834000 999000 1230800 90 735000 777000 1396000 100 596000 726000 1280000

Now, I made a table for 70 years, with each type of tree and with possible combinations, with the number of years left and the profit obtained.

 Sitka Spruce European Larch Lodgepole Pine Profit (£) 30 40 0 697200 30 0 40 574400 40 30 40 418400 0 40 30 451700 70 0 0 1056800 0 70 0 1098000 0 0 70 1064000 30+40 0 0 731000 0 30+40 0 541200 0 0 30+40 328900

The manager should plant the European Larch for 70 years, obtaining a profit of £1098000 per hectare.

I made a similar table for 90 years:

 Sitka Spruce European Larch Lodgepole Pine Profit (£) 90 0 0 735000 0 90 0 777000 0 0 90 1396000 30+30+30 0 0 864000 0 30+30+30 0 396000 0 0 30+30+30 127500 40+50 0 0 1066000 0 40+50 0 1207000 0 0 40+50 852400 60+30 0 0 1052000 0 60+30 0 1256000 0 0 60+30 912700 30 30 30 426500 40 50 0 1241000 40 0 50 1009000 50 40 0 1032200 0 40 50 975000 0 50 40 1084400 50 0 40 909400 60 30 0 896000 60 0 30 806500 0 60 30 1166500 30 60 0 1412000 0 30 60 1002200 30 0 60 1158200

The manager should plant the Sitka Spruce for 30 years, and then the European Larch for 60 years.