Climb the ladder or leap straight to the top
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Every step takes a turn, and so does going straight to the top. Nothing here is tied to one game: any ladder you can climb one rung at a time or leap in a single move is the same arithmetic.
Results
| Cost of climbing all the way from the bottom | — |
| Cost of leaping from the bottom | — |
| What leaping costs on top of climbing | — |
| What each turn saved costs, from the bottom | — |
| First step from which leaping costs no more | — |
From each step, what the two ways cost
| Steps already climbed | Cost of climbing the rest | Turns that takes | Cost of leaping from here | What each turn saved costs |
|---|
The last column is the one worth reading, and it usually rises as you go down the table, which is the opposite of what it feels like. Leaping from the bottom buys two turns at once and spreads its premium over both of them; leaping when you are nearly there buys a single turn and pays the whole premium for it.
So rushing is at its cheapest per turn when you are furthest behind, and at its most expensive when you are almost home. The instinct to climb patiently early and then reach for the shortcut at the end has it exactly backwards on price, whatever else may be said for it.
What the page cannot price is what those turns are worth to you, and that is where the real decision lives. A turn early in a game that you are winning is worth little; a turn in a race you are losing by one can be worth any premium at all.
Which is actually cheaper, climbing or leaping?
Climbing almost always costs less in total, because the steps are priced to be affordable one at a time. What leaping buys is turns, and the page prices those turns instead of arguing about the totals.
With steps of two, three and four against a leap of twelve, climbing from the bottom costs nine and takes three turns, while leaping costs twelve and takes one.
When is rushing at its cheapest?
When you are furthest behind, which is the opposite of how it feels. Leaping from the bottom buys two turns at once and spreads its premium across both; leaping when you are nearly there buys one turn and pays the whole premium for it.
In the example below the same three-point premium works out at one and a half per turn from the bottom and three per turn one step up.
| Steps already climbed | Cost of climbing the rest | Turns that takes | Cost of leaping | Price of each turn saved |
|---|---|---|---|---|
| 0 | 9,00 | 3 | 12,00 | 1,5000 |
| 1 | 7,00 | 2 | 10,00 | 3,0000 |
| 2 | 4,00 | 1 | 7,00 | — |
Why is the last row blank?
Because from there both ways take a single turn, so leaping saves no time at all and there is no price per turn to quote. Putting a number there would be inventing one.
What that row does show is the premium in plain terms: paying three more for nothing but a different route.
Does this tell me whether to rush?
It tells you what a turn costs, which is the half of the question that can be answered exactly. What a turn is worth depends on the game in front of you and cannot be read off any table.
A turn early in a game you are winning is worth little; a turn in a race you are losing by one can be worth any premium at all.
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